[P2 paradigm/exact] Scheme-theoretic operator state: unify Q-adic Gram/Jantzen layers with Hankel multiplication algebras

[LightChainr]

2026-08-31 本轮固定干预已完成；降为P2

结果、代码与可复现实验包已随Draft #509推送，尚未合并。

在Huawei ZyTrST完成width8、η=0,±1/4的固定干预，保留原读出和八个lags；完整1430态模型使用联合186维精确扇区，每个η重新求平稳分布。实际计算1.709秒，无MC、扩宽或参数扫描。

• η=1/4、lag=.5时，旧U=C(0)^−1C(t)给出U−+=−.00889715、U+−=−.01747217，真实cross传播不能由共同时间换元解释。
• 固定14→16个配置函数使最大C0尺度误差3.70804e−4→8.51022e−5（4.36倍）；弱minus的lag4相对误差仍−12.64%，改善但未闭合。
• Galerkin比较使用各η完整模型的数值平稳分布，不能称为只靠η=0矩阵的独立预测。±1/4有限对比不是η=0精确导数；零频积分本轮未计算。

本轮有限干预交付结题，Issue保留开放以保存尚未解决的一般理论问题。 目前降为P2，不继续默认占机、扩width10、加mark或扫描。下一次调用需对应具体square-site源/读出映射，或一个确实需要精确切线的理论问题。完成结果、负结果及余项分别保存，不再派首次干预。

以下原交接与讨论正文完整保留，旧“首次待办”按上方结果更新。

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2026-08-31 P1团队交接：一次固定速率干预

本轮作为有界并行验证，建议约10%的团队注意力，不阻塞 #154 / #334。T4第四阶桥和慢极点/权重分解已经完成，不再派首次解释或扩Krylov层。

首份交付：固定总尝试率的G_eta=(1+eta)Σ(J−I)+(1−eta)Σ(D−I)，保留原配置函数和lags，计算零点cross-ray切线/零频积分，以及eta=±1/4下完整正模型与旧14/16维观察子空间的响应对照。需要联合186维扇区，同时包含平稳测度和传播子导数。

明确预测：K交换join/detach，使同ray响应对eta偶、cross-ray响应对eta奇；同ray一阶零作为对照。cross-ray切线若非零，纯时间重标不能解释；若得到额外精确零，交付选择规则；旧观察量若预测失败，交付反例。

一次干预之后，未建立到square-site实际源/读出的映射，就不默认扩width10、加mark、扫速率或拟合指数。当前width4/8的i^j保持波长4，本次是参数外推，不称固定k尺度验证。结果与下一判断回写本Issue。团队计划。以下保留原文。

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Research delivery — 2026-08-31 e / completed million-sample response and physical next readout

新结果已完整读入：P398 current-source geometry与科学边界，固定source commit 33c6028，branch_only。

继已完成current deletion后，这次(e,Je)读出已经保留全部瞬时流方向，但plus慢质量仍为3.62846对1.95575，t=4相关仍丢失99.7429%。在G与reversible S中，遗漏的初始曲率都精确等于 ||QSe||²（minus .0602551994、plus 2.9272489452）。Je的Cauchy最优性是构造恒等式，不是新发现的最佳物理场。

下一问题因而移到明确的hidden reversible geometry/transport及microscopic/matching overlap；首次current删除和首次Je observer均已完成，不再重新派发。同一个1430态对象的多种投影不是独立实验，投影observer pair也不是新的两态Markov链。当前交接。本次未重跑谱、测试或服务器。

Previous body preserved — historical context

Research delivery — 2026-08-31 d / completed physical source and updated handoffs

520a9d21的stationary-current删除实验已完成：在相同1430态、pi、source和exit rates下，用S=(G+G*)/2移除全部稳态概率流，fast/slow反转仍存在，crossing仅从.265657到.272263（2.49%）。

这说明此有限例中普通reversible正混合已足够，非实mode、negative residue或Jordan并非必要；不说明S一定是局部square-bond transfer word，更不命名square-site Matching能量场。与width8 memory/motifs是同一个exact对象，不是独立replication。

因此下一项不能再写“首次current deletion”。请用具体microscopic/matching overlap或物理rate/geometry预测推进；该分支已有后续head，派具体读出前先读对应新结果，避免撞车。唯一下一步表。本次未重跑谱、test或服务器。

Previous body preserved — historical context

Context reconciliation — 2026-08-31 c / completed work before next assignments

本轮补读发现原正文仍停在width8 protected rays，后续39e0660→c9dc218的memory/motifs已完成。

• normalized plus反馈6.81225倍＝inverse source variance12.4744倍×bare feedback.546099倍：主要是近暗源放大，不是plus裸耦合更强。
• R/T2在每条protected ray内等价，不是两份独立机制证据。triplet边界邻接、join/detach有向预算和GT_m的size/contact hierarchy已经给出；首次hidden force或triplet分析不再待做。
• 可以据此提出改变rate/geometry的物理响应预测，但rate变化还会改变stationary law与projection，现有预算不是该因果导数。width8连续fixed-i、width5离散C5和原square-site Matching保持不同对象；detach是frontier singleton chipping，不是任意bulk删点。

全项目交接将这些已完成输入接回#333/#154/#275，尚未宣称能量身份、连续场数或通用低维闭合。本轮只整理，不重复算谱、跑测试或派新任务。

Previous body preserved — dated historical context

Scientific output update — 2026-08-31 second continuation

Draft #267 / 4846adf 已完成固定读出分析，只消费已有width5 C(d)/residues。

仅由等时C0定义 J=L−C_LA(0)/C_AA(0) A，不看非零lag选择系数：d1单位方差信号.020216，两慢模遗漏77.98%，对照AA .077%、LL1.976%、整矩阵.900%。J第三模近距离可见；远距离抵消时必须同时报绝对信号。额外slow-null滤波的100%遗漏是构造结果，不是独立否证。

另已读 branch_only 552c45d width8结果：连续fixed-i两Kreweras rays各93维，微观泄漏GL=−3L+T2、GA=−3A+R。它不是width5离散C5模型。下一步用这些具名发射量/对偶预测跨宽度或微观运输；firstfixedreadout、firstwidth8已完成。精确状态、观察者有效慢模和连续场身份分别记录。

Previous body preserved — historical context

2026-08-31 物理接口与宽度延拓均已有结果

• 正权物理两点矩阵已经完成。 branch_only e38fe76 给出width4真实AP/landing C(d)及两个非零普通本征值；b35e100完成完整正权h/v族；dbd4081给出同宽连续距离fingerprint。旧正文“物理矩阵尚未产出”不再是当前状态。
• 新 Draft docs: recover scientific frontier and score production mechanisms #267 / 8f7a587 已把同一adjacent-pair/singleton微观指标与first cyclic character延拓至width5，正权42态、h=v=1/2。实际正间距两点Hankel精确秩8，传播多项式无重根；这是有限圆柱传播，不是8个CFT场。精确结果。
• 消费该矩阵得到一个更有用的区别：两最慢模不拟合residue，整矩阵相对误差在d1/d2/d4为 .8996%/.1523%/.004050%；但 U=C0^-1C(d) 的 U2−U1² 相对缺口7.8838%。精确状态、有效慢模与原两读出的自主闭合不是同一个概念；whole42state过程仍是Markov。数值分解。
• 下一步已越过首次width4/width5计算：比较具名读出在明确距离上的慢模截断误差及未被两模保留的方向；full-Q/连续极限身份仍是不同问题，不再重复自动jet闭合。

当前注意力顺序与本轮科学进展消费这些完成结果，不另设准备阶梯。未运行测试套件、未用服务器、未增MC样本；优先级不是许可，不关闭/锁定任务、不自动合并PR。

Earlier proposal and context review — historical text preserved

2026-08-31 上下文恢复（原提案保留在下）

• rooted/landing 接口已完成一大步：5389200（branch_only）闭合既有九 mark、23维 accumulator，radical 维数4；不再是“下一步再造同一 rooted closure”。
• afc619c 已完成 second jet：唯一 X2=0，Jantzen valuations 为0^19,1^4，J2=0；同一 fixed-radical projected-Gram 的更高阶约束自动成立，重复这些阶数不会新增识别信息。
• 正权物理观测仍未做成。 既有两个 charged readout 在明确正权 width-4 strip 上的 charge-neutral 2×2 连通响应矩阵尚未产出；unprojected Gram、物理 emission 与 all-Q transfer/Jordan 实现也未建立。可直接消费现有模块，而非继续累积同型 closure。
• P250 的 rank8 截断/非平坦问题仍与本 Q-adic 模块分开；两边有 nilpotent 语言不构成同一物理机制的证据。
• 当前总览与下一步见 Draft docs: recover scientific frontier and score production mechanisms #267 的上下文恢复与注意力顺序；原提案全文保留在下。

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Trigger

Two late frontiers now look like different coordinate views of the same algebraic problem.

Generic-Q / Gram side

The #333/#370 exact program has moved far beyond a generic “Jordan fit”:

• the join-only width-3/4 closure has an exact first-jet Gram/source obstruction;
• adding the ordinary scalar FK detach/loop weight does not open the missing radical direction;
• one genuine terminal mark has nonzero Q-velocity on the extended radical but is still insufficient;
• the minimum falling-factorial scalar-mark families tested at widths 3/4 remain incompatible with the first-jet Gram form, and the width-4 scalar family is exhausted.

So the unresolved object is not simply “one more scalar coordinate”. The obstruction lives in the degeneration structure of the bilinear form itself.

Context-Hankel / charged side

The P250 endpoint-Hankel program has also crossed a qualitative boundary:

• within each hand, rank <= 5/6/7 is statistically eliminated in the frozen radius-6 block;
• rank 8 is only the first compatible truncated rank, not an exact or flat physical dimension;
• the old Alexander-R2 degree-two null line does not extend to equality of the complete degree-three rank-eight kernel planes;
• endpoint data still do not determine ordered T_x T_y versus T_y T_x structure.

PR #385 adds the identifiability warning: an ordinary diagonalizable family can approach a Jordan semigroup arbitrarily closely at finite noise. Therefore “more precise eigenvalue fitting” cannot by itself separate reduced ordinary states from a confluent/non-semisimple limit.

I think the natural next object is scheme-theoretic rather than spectral.

Literature collision

Two recent algebraic developments are unusually close to what the repository is already computing.

1. Martin--Senécal--Spencer, arXiv:2601.17445 (Cell modules for the Temperley-Lieb algebra in mixed characteristic) completely describe TL cell-module submodule structure and explicitly investigate two-dimensional Jantzen-like filtrations.

2. Hernández Caro--Ryom-Hansen, arXiv:2210.03847 diagonalize cell-module bilinear forms in a TL/blob/Soergel setting and use their vanishing structure to construct Jantzen-type filtrations and a graded sum formula.

3. Bernardi--Jelisiejew--Reig Fité, arXiv:2606.30600 prove that Hankel flat-extension completion is equivalent to completing the multiplication tensor of an Artinian Gorenstein algebra. Unknown moments and unknown multiplication-tensor coefficients are the same variables in two coordinate systems; the resulting multiplication matrices and commutation equations coincide.

4. Ikhlef--Morin-Duchesne, arXiv:2312.14837 give a useful periodic-TL control where non-generic standard modules are reducible indecomposables with a radical and quotient, and relate their fusion to logarithmic bulk CFT connectivity operators.

These suggest replacing two ad hoc languages—“try another marked Q-direction” and “fit another Hankel root”—by canonical filtrations and finite algebras.

Core proposal A — compute the Q-adic/Jantzen filtration instead of guessing marks

Let

t = Q-1

and let W be one declared generic-Q connectivity/closure module over a local coefficient ring near t=0, with invariant bilinear form

G(t) = G0 + t G1 + t^2 G2 + ... .

The current repository mostly studies rad(G0) and the first-jet compatibility equation. Instead define the higher vanishing filtration schematically by

J^k = {
  v mod t : <v,W>_t is divisible by t^k
}.

The exact implementation need not assume a theorem from the papers. Compute it directly from the repository's polynomial Gram matrix.

A practical exact route is:

1. construct G(t) symbolically to sufficient t-order;
2. compute Smith/invariant-factor data over Q[t] localized at (t),
   or an equivalent exact t-adic elimination;
3. record the valuations a_i in elementary divisors t^(a_i);
4. recover dim J^k/J^(k+1) and canonical basis representatives;
5. act with join, detach, translation/seam and declared marked morphisms on the associated graded pieces.

This is a stronger object than rank(G0) or one first derivative. It answers how many independent confluent directions the degenerating form actually supplies, at which Q-order, and in which representation sector.

Immediate use on the current negative results

Project the already-tested scalar/falling-factorial marks into the canonical layers

J^1/J^2,
J^2/J^3,
...

instead of asking whether an arbitrary finite list closes the first-jet equation.

My guess is that the scalar marks fail because they do not span the first required associated-graded representation. If the missing layer is intrinsically rooted/connectivity-valued, the filtration should show this before another mark family is invented.

Core proposal B — interpret P250 flat extension as a finite algebra, not “r states”

Let the mixed-displacement endpoint moments define a truncated functional L and Hankel matrix H_d.

If a rank-r flat extension exists, the Bernardi--Jelisiejew--Reig Fité result says the correct reconstructed object is a finite quotient algebra schematically

A = C[x,y]/I,
dim A = r,

with multiplication operators

M_x, M_y : A -> A.

The important distinction is then not merely

rank = 8 or rank = 9?

but

reduced finite scheme       <-> distinct ordinary support states,
nonreduced local components <-> multiplicity / nilpotent directions,
                              hence confluent/Jordan-like memory.

This is exactly the distinction that finite-noise eigenvalue fits cannot robustly resolve in PR #385.

Concrete P250 analysis

Using the locked radius-6 moments, separately for plus and minus hands:

1. attempt the next-degree flat extension without imposing the rejected full R2 kernel-plane equality;
2. if compatible, reconstruct the annihilator ideal / multiplication tensor in a basis selected by invariant Hankel structure rather than fitted exponential roots;
3. compute Hilbert function, multiplication minimal polynomials, nilradical dimension, and Jordan type of generic linear combinations a M_x+b M_y;
4. distinguish “eight reduced points” from schemes with multiplicity only when the exact/structural information supports that distinction;
5. treat the upcoming ordered connected two-morphism rectangle as an extension beyond this commutative endpoint shadow: it asks whether the physical category action factors through the reconstructed commutative algebra or needs ordered/path memory.

If flatness fails at rank 8, that is itself cleaner than adding another pole: rank 8 was only a truncated Hilbert-function lower bound, not a realizable finite algebra.

Core proposal C — compare invariants across the two constructions

Do not identify the P333 Q-adic module and the P250 spatial moment algebra merely because both contain nilpotents.

Instead compare invariant summaries:

Q-adic side:
  Jantzen-layer dimensions,
  radical/Loewy length,
  representation characters,
  nilpotent action on associated graded pieces;

Hankel side:
  Hilbert function,
  reduced/nonreduced support type,
  nilradical dimension,
  Jordan type of multiplication operators,
  sector/deck/C4 action on the finite algebra.

A genuine common mechanism should reproduce more than one scalar log coefficient: it should give compatible extension geometry in parameter space and context space.

Strong conjecture

The minimal predictive object in Matching One is not a list of continuum fields/eigenvalues. It is a finite (possibly nonreduced) operator-state scheme together with typed module actions. Jordan behaviour is then local nilpotent geometry of this scheme, while ordinary multiple fields are its reduced support.

In this picture:

P333 Q->1 degeneration
    = tangent/Jantzen geometry of the module family;

P250 finite Hankel realization
    = coordinate algebra of the observable/context scheme;

PR #385 near-coalescing ambiguity
    = reduced points approaching a nonreduced point in the same closure;

PR #393 common-nilpotent semigroup
    = one very small formal local-algebra model, not yet the full scheme.

This also explains why increasing a scalar state count can be the wrong reaction to every failed closure: the new information may be a multiplicity/extension coordinate, not a new reduced point.

First exact controls

1. TL cell-module control

On a tiny Temperley--Lieb module where the radical/submodule structure is already known, compute the same t-adic Gram filtration using repository-style exact arithmetic and compare with the known radical/quotient structure.

2. Synthetic Hankel controls

Construct exact moment tables for:

A. r distinct reduced points;
B. one double point + r-2 reduced points;
C. a length-3 local algebra;
D. two near-coalescing but still reduced points.

Verify that the flat-extension/multiplication reconstruction separates A/B/C exactly while D approaches B continuously, reproducing the PR #385 identifiability boundary.

3. Current P333 width-3/4 data

Compute the Q-adic invariant factors before adding any further scalar marks. Compare the tested mark families with the canonical associated-graded layers.

4. Current P250 endpoint moments

Treat rank 8 as a candidate truncated algebra dimension, not a field count. Attempt flat/multiplication completion and publish the scheme invariants or the exact obstruction to such a completion.

Boundaries

• The current connectivity module has not yet been proved to be the particular TL/blob cell module of the cited Jantzen literature. The literature motivates the invariant; the repository should compute its own filtration exactly.
• Artinian-Gorenstein/Hankel reconstruction applies only when the required flat-extension/moment hypotheses are satisfied. A statistical rank lower bound alone does not create such an algebra.
• At finite noise, reduced versus arbitrarily near-nonreduced alternatives remain nonseparable without structural constraints, exactly as PR Exact identifiability controls: Jordan closure and a C4-protected five-state quotient #385 warns. Use exact controls and declared symmetry/algebra constraints for sharp claims.
• A nonreduced finite algebra is not by itself a Virasoro Jordan module or an LCFT field identification.
• Every derived coordinate from the same raw block remains one correlated evidence block.

Why this may simplify the repository

This proposal potentially replaces two expanding search trees

more Q-dependent marks,
more fitted spatial eigenmodes,

by two canonical algebraic constructions

Q-adic radical/Jantzen filtration,
Hankel multiplication algebra / finite scheme.

The output is structural and proof-carrying: a filtration, an ideal, multiplication tables, radical dimensions and representation actions. Those objects survive basis changes and are much closer to the non-semisimple algebra that the current data are already forcing us to confront.

Related: #218, #249, #250, #333, #370, PR #382, PR #385, PR #393.

[LightChainr]

Immediate refinement from concurrent #400: the scheme is generator-relative, not a universal interpretation of Hankel rank

#400 appeared concurrently with this issue and supplies an important correction to Proposal B.

A single critical power law sampled on additive lattice translations can have full Hankel rank at every order, while the same field sampled on a geometric/log-radius grid becomes a rank-one exponential sequence; one logarithmic partner becomes rank at most two on that dilation grid. Therefore

additive translation Hankel rank
!= scaling-field count

in general.

This should be built into the scheme language rather than treated as an external caveat.

Refined statement

A finite Artinian/Hankel multiplication algebra is a model of a declared finite constant-coefficient context realization. It is not automatically the physical operator-state space.

Every use of the scheme reconstruction should therefore type the generator first:

spatial lattice translation,
log-radius / dilation,
Gaussian-cover composition,
deck action,
occupation-rank evolution,
ordered intervention/morphism.

The same continuum field may have an infinite-dimensional additive-translation descendant orbit but a very small dilation/Gaussian-semigroup realization.

Consequence for P250

The radius-6 endpoint rank-eight result should be split into two questions:

1. Kinematic question: does the observed displacement axis belong to a finite polynomial-exponential/Artinian class at all, or is the rank growth already explained by a critical translation kernel? [P2 analysis/control] Separate spatial-translation Hankel rank from scaling-field rank before enlarging P250 #400 is the right control for this.
2. Conditional algebra question: if a finite constant-shift realization remains scientifically justified for a declared context, what finite scheme/multiplication algebra realizes it, and is that scheme reduced or nonreduced?

So the #398 flat-extension analysis should not automatically commission a higher rank because rank 8 is first compatible. It should use #400's coordinate/generator calibration first and apply Artinian reconstruction only to contexts where finite closure is a meaningful hypothesis.

Where the scheme picture remains especially natural

The scheme language is still compelling for

Gaussian multiplier composition,
log-radius transfer,
finite deck/twist actions,
finite exact generic-Q module quotients,

because these are algebraic/semigroup contexts rather than raw additive spatial translation.

A stronger long-term object may therefore be a typed functor whose different generators have different realization categories:

dilation/cover generator -> finite scheme / local algebra when supported,
spatial translation      -> conformal/torus kernel, possibly infinite descendant rank,
ordered morphisms        -> noncommutative/category representation.

This refinement makes #398 and #400 complementary rather than competing. The scheme is the finite algebraic component of the context category, not a universal replacement for CFT kinematics.

[LightChainr]

Exact refinement: distinguish base-parameter nilpotence from a fixed-Q physical extension

I read this issue and its #400 generator-relative refinement after the width-3/4 scalar-mark exclusions 081a5ed / e7e6c80. I will not open a duplicate Jantzen/Hankel program. One inexpensive exact control should be added before interpreting a Q-adic nonreduced object.

1. Every differentiable matrix family has a formal jet, even without a physical Jordan block

Put t=Q-1 and work modulo t^2. For A(t)=A0+t A1, its action on (v0,v1) is

J(A) = [[A0, 0],
        [A1, A0]],
N    = [[0, 0],
        [I, 0]].

Then exactly

J(AB)=J(A)J(B),  N^2=0,  [N,J(A)]=0.

This common nilpotent exists for completely ordinary scalar families. It is therefore a bookkeeping property of the parameter-thickened family, not evidence for a nonsemisimple physical operator at fixed Q.

2. A compatible jet Gram form is also automatic under the correct assumptions

Suppose G(t)=G0+t G1 is symmetric and A(t)^T G(t)=G(t) A(t) through first order. Taking the coefficient of t in the pairing gives

B = [[G1, G0],
     [G0,  0]],
J(A)^T B = B J(A).

The upper-left block is precisely the differentiated Gram equation; the off-diagonal blocks are the zeroth-order equation. For a nontrivial adjoint/diagram involution, use the corresponding starred generator, not an unproved self-adjointness convention.

This construction is a positive control for differentiation and bilinear-form bookkeeping. It does not solve the repository's constrained same-module/endpoint/radical obstruction: it changes the module, and it may retain a nonzero parameter action.

3. The two-by-two counterexample that should be frozen

Take the scalar family D(Q)=Q, G(Q)=1, which satisfies D(Q)^2=Q D(Q). Its jet is

J(D) = [[1,0],[1,1]],
B    = [[0,1],[1,0]],
N    = [[0,0],[1,0]].

It has a Jordan block and is B-self-adjoint. Nevertheless

J(D)^2 - J(D) = N != 0,
J(D)^2 = (I+N) J(D).

Thus it represents the parameter-thickened relation, not the Q=1 relation D^2=D. Declaring it a physical Q=1 logarithmic module would be a category error. In this toy example quotienting by the parameter action destroys the apparent extra Jordan direction.

4. A sharper gate for the scheme program

Label every nilpotent by its source:

base-ring nilpotent t,
radical/extension at fixed Q=1,
context multiplication nilpotent,
physical dilation/transfer nilpotent.

Before identifying any two, require an explicit module/intertwiner diagram and verify the specialized algebra relations, source/readout maps and Gram pairings in the actual quotient. In particular, ask whether a rooted-connectivity extension supplies a nontrivial fixed-Q extension after the base-parameter nilpotent acts trivially; a formal dual-number enlargement by itself does not.

This gives a constructive test rather than only a warning: run the same relation/Gram verifier first on the scalar toy (must reject physical-Q=1 status), then on a known genuine indecomposable periodic-TL module, and only then on the proposed new connectivity quotient. A failure can be localized as a parameter-specialization obstruction, an endpoint/Gram obstruction, or a true extension obstruction.

Literature addition

Giaquinto--Mastnak, arXiv:2607.12247, submitted 14 July 2026, develops an ordinary Temperley--Lieb center calculation using cellular filtration, radicals, deformation and leading Gram terms. It is a useful additional exact control, but its ordinary TL result should not be imported as a periodic-TL/Potts theorem without the dictionary. Ikhlef--Morin-Duchesne, arXiv:2312.14837, provides the more directly relevant periodic-TL irreducible/fusion setting.

The point is not that formal jets are useless. They are the right baseline for parameter response. The nontrivial research target is what remains as a typed physical extension after specialization, not the existence of the automatically manufactured nilpotent. This also separates #398's parameter-family geometry from #401's occupation-process continuation state; neither finite rank nor nilpotence alone identifies the two.

[LightChainr]

Scientific card: canonical Q-adic/Jantzen control

• Mechanism space changed: the base connectivity Gram degeneration has no hidden higher Q-adic layer at widths three or four: the valuations are 0^1,1^4 and 0^1,1^13, with J2=0 exactly.
• Width-three projection: the frozen scalar and C3 charge-one responses span all 4/4 grade-one dimensions.
• Width-four projection: the frozen scalar/charge-one/charge-two responses span only 6/13; uncovered sector dimensions are trivial 2, charge one 2, charge two 3.
• New discriminator: the missing width-four information is multiplicity/rooted-connectivity inside already present C4 sectors, not another terminal irrep or a deeper power of Q-1.
• Nilpotent type: the square-zero dual-number action is base-parameter bookkeeping and vanishes on fixed-Q specialization; a physical or fixed-Q extension still needs its own module and closure certificate.
• Not proved: no LCFT field, physical transfer Jordan block or periodic-TL cell-module identification.

[LightChainr]

Scientific card: minimal rooted completion of width-four gr1

• Mechanism space changed: the exact 2+2+3 width-four grade-one deficits are all filled by a seven-coordinate AP/OP/DP rooted source-to-landing registry; no new terminal character is introduced.
• Exact result: old-plus-rooted rank is 13/13; the square B-coordinate matrix has determinant 3072, and every C4 translation residual is zero.
• Minimality: seven is both the sector-deficit lower bound and the number attained, so this is a minimal linear associated-grade completion.
• Scientific meaning: the missing information is connectivity multiplicity within the trivial, charge-one and charge-two sectors, not a missing C4 irrep.
• Not proved: this span certificate does not yet show that the corresponding extended module satisfies affine/endpoint/radical/Gram/source closure.
• Next discriminating object: build exactly this frozen rooted registry as a module and run the closure intersection; do not add more response coordinates first.

[LightChainr]

Scientific card — minimal coupled rooted closure

Committed and pushed 5389200 on theory/p398-rooted-module-closure-20260830; no PR or merge. Exact report.

• New result: the seven rooted derivative marks alone give an empty inherited P333 affine/endpoint/radical-Gram/source intersection. Coupling them to the already-defined C4 charge-one landing doublet makes that intersection uniquely nonempty: X0=diag(T,R), V=0.
• Exact certificate: rooted7 has dim W=21, rank G0=15, radical=6, Gram-skew rank=2. Rooted7+old2 has dim W=23, rank G0=19, radical=4, Gram-skew rank=0. The remaining four-dimensional leading radical Gram is nondegenerate, so success does not empty the tested space.
• Why the entire intersection is decided: ordinary shifted-Hom rank is 195/196; common-left-invariant rank is 13/14. Source/mark transport forces the canonical point uniquely. The rooted7 obstruction has coefficient rank 0, augmented rank 1 and an explicit pure-charge-one radical-vector witness.
• Minimum repair: one rational C4 charge-one doublet (two coordinates). No new terminal character: the old doublet and rooted7 each fail alone but close when sharing the ordinary Gram/source block.
• Semantics: retained derivative-response accumulators, not an active seven-state connectivity submodule. Active reference semantics separately saturates 7 -> 13 -> 14, with a one-coordinate J0 AP_sum witness.
• Observer/source/dependency: width-four exact connectivity/Gram algebra, C4 multiplicity sectors, all-singleton source and exact mark transport; the same P333/P398 dependency block, not independent numerical evidence.
• Not proved: unprojected Gram self-adjointness (explicit residuals remain), all-Q closure, physical transfer realization, continuum field/Jordan identity. Coupling here is through the declared common Gram/source block, not a measured physical exchange.
• Next discriminator: a physical coupled emission/transport or all-Q extension of this nine-mark lift; no further terminal-character scan.

Six focused tests pass. Zero new Monte Carlo.

[LightChainr]

Scientific card — second jet survives, but the projected gate is saturated

Pushed afc619c, branch theory/p398-rooted-second-jet-20260831, parent 5389200; no PR/merge. Exact report.

• Family audit: the existing retained family explicitly has constant emission (Q-1)E_i and derivative-reference Gram columns. With the nine frozen references and E_i=C0^T P_i, the second detach coefficient is zero and the second Gram coefficient is fixed. No extra observable or changed Gram convention was introduced.
• Second jet: X2=0 uniquely. Filtration/mark transport fix 207 of 529 entries; the remaining 322 have a sufficient exact homogeneous subsystem of rank 312 plus 10 independent source constraints. The second projected Gram residual is exactly zero.
• No next Jantzen layer: valuations are 0^19,1^4; dim J2=0. The leading four-dimensional radical Gram determinant is 14.
• New structural conclusion: after removing charge one, the remaining radical is three trivial plus one sign direction. Its induced translation satisfies R_B^2=I. Together with C4 covariance this forces H(Q)R_B=R_B^T H(Q) for the entire polynomial. All five Taylor coefficient checks (degree 0–4) vanish exactly.
• Consequence for research allocation: second-jet survival is genuine, but higher repetitions of this same fixed-radical Gram gate cannot supply new operator-identification evidence. The next discriminator must involve a genuinely different quotient/physical-emission constraint.
• Observer/source/dependency: same width-four nine-mark retained derivative family, fixed ordinary source and mark transport; same P333/P398 exact dependency block.
• Boundary: this is not unprojected Gram closure or a physical transfer/Jordan realization. The known stronger Gram failure was neither rerun nor repaired.

Three focused tests pass; zero new simulation.