[P1 exact/paradigm] Intrinsic ambient-homology source: generate Matching One without a Potts-Q lift

[LightChainr]

2026-08-31 上下文恢复（原提案保留在下）

• intrinsic rank-source 与 canonical A_top/E_top、出生 current/risk-hazard 已有精确及生产结果；额外 unmarked source 导数受 X³=X 限制。integral saturation c1a72e5 在该 branch_only 定理中给出 iota=1，使 index≠1 不再提供额外状态自由度，但不消除 line/local/history 信息。
• source-response 的正分离已经测到。 F3 两行中一行是 symmetry-null；后续 F5 两条 D4-even readout（2d2a9ab，N325/N425各20k）分辨 W_line/JS 的不同响应方向，joint chi²_4=149.93/246.93。旧 O_far/O_sep4 未分辨不能再解释为 source equality。
• 这与 canonical matching 的载荷是不同问题。PR Score real same-stream P439 crosswalk and unresolved M loading #451 bfbceb2 的 M=0 仍相容（p=.58155）；Draft 8498d62 direct/plateau 两分量也都弱，joint zero4.69005/8、p=.79013，不支持“大项抵消已解释零 M”。
• 尚需把已定义的 F5/line-typed readout 运到可区分的几何并测其 M loading；不能用更多 K_A-only 尺寸、重复 F3/F5 首测或把同一 filtration 称为独立微观数据源来替代。
• 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。

---

Trigger

The latest exact work changes what should be regarded as the primitive object.

Digital Alexander duality gives, on the honest square-cell torus,

r(omega) = rank im[H1(K(omega)) -> H1(T^2)] in {0,1,2},
M_N(p)   = E_p[r]-1 = P_2-P_0.

The threshold-rank archive has also been reinterpreted pathwise:

K_minus = first essential ambient-H1 birth,
K_plus  = second essential ambient-H1 birth.

At the same time #333 has now proved an important negative/clarifying statement. Two natural generic-Potts lifts of the same Q=1 observable,

H_Q = W_2D-W_0D,
C_Q = W_2D-Q W_0D,

agree at Q=1 but have different Q tangents; on the exact critical-polynomial section C_Q is identically zero while the unweighted homology lift has the nonzero tangent pi_0. Thus the raw Q tangent is a property of a declared generic-Q lift, not of Matching One alone.

There is a more intrinsic parameter direction already present at Q=1 and on the lattice itself: couple an auxiliary source directly to the ambient homology rank.

Canonical topological source

Define the centered ambient-rank variable

X(omega) = r(omega)-1 in {-1,0,+1}

and its finite-volume generating function

Z_top,G(p,s)
  = E_p[ exp(s X) ]
  = P_0(p) exp(-s) + P_1(p) + P_2(p) exp(+s).

Equivalently, with u=exp(s),

Z_top,G(p,u)=P_0 u^-1 + P_1 + P_2 u.

This definition:

• exists directly for site percolation at Q=1;
• requires no generic-Potts continuation;
• requires no field-normalization choice;
• uses exactly the ambient-homology observable already proved by the repository.

At s=0, Z_top=1, so

partial_s log Z_top |_(s=0)
 = E[X]
 = M_N(p),

partial_s^2 log Z_top |_(s=0)
 = Var(r).

The mixed thermal/source derivative is equally canonical:

partial_p partial_s log Z_top |_(s=0)
 = d/dp E[r]
 = M_N'(p),

which is the total rank-birth pivotal susceptibility from #276.

So Matching One is exactly the linear response to an intrinsic topological source.

Exact two-dimensional source algebra

Because X takes only -1,0,+1, configuration by configuration

X^3=X.

Hence the algebra generated by the unmarked ambient-rank observable is

R[X]/(X^3-X).

After removing the constant, it has exactly two canonical directions:

odd:   X
       -> E[X]   = P_2-P_0 = A_top = Matching One,

even: X^2
       -> E[X^2] = P_2+P_0 = 1-P_1 = E_top.

In particular

partial_s^(2k+1) Z_top|_0 = A_top,
partial_s^(2k)   Z_top|_0 = E_top   for k>=1.

This is an exact minimal topological state, not a learned PCA/Hankel rank.

It gives a sharp boundary for the low-rank discussion: any additional scalar direction required by the thermal jet cannot come from unmarked ambient rank alone. It must enter through thermal/history dependence, projective-line or integral-subgroup marks, local/defect data, or another bulk field.

Do not identify this exact 2D source algebra with a rank-2 Virasoro/Jordan module. The RG action on these two observables is a separate question.

Full Alexander functional duality

For the primal square-site configuration and its matching complement, the exact theorem gives

r_G(omega) + r_Ghat(omega^c) = 2,

so

X_G(omega) = -X_Ghat(omega^c).

Because complement sends Bernoulli p to Bernoulli 1-p, the whole generating function obeys

boxed:
Z_top,G(p,s)
 = Z_top,Ghat(1-p,-s).

This is stronger than the first-moment matching identity. It implies the complete cumulant hierarchy

kappa_n^G(p)
 = (-1)^n kappa_n^Ghat(1-p)

for X wherever the digital-Alexander theorem applies.

The ordinary matching function is just the n=1 member.

On an exactly self-matching realization at p=1/2, Z_top(s) must be even and every odd topological-source cumulant vanishes exactly. This is a cheap positive control.

Continuum interpretation without a Q lift

Arguin's critical Q=1 torus relation gives

pi_2(tau)=pi_0(tau)

for every modulus. Therefore the continuum source function

Z_top,cont(tau,s)
 = pi_0(tau)e^-s + pi_1(tau) + pi_2(tau)e^s

is exactly even in s at criticality:

Z_top,cont(tau,s)=Z_top,cont(tau,-s),
partial_s Z_top,cont(tau,0)=0.

The finite square-site matching signal can therefore be restated as:

the lattice approaches an exact continuum rank-source reflection symmetry, and Matching One measures its leading finite-spacing odd defect.

This formulation does not decide which irrelevant singlet produces the defect. The current vacuum/KdV, thermal-Q4/Jordan, ordinary mixing and topological/cover alternatives remain separate candidates.

But it provides a canonical source with respect to which odd/even is exact and lift-independent.

Phase A — exact source oracle on current HNF frontier

Use the existing digital-Alexander state cache / HNF filtration machinery; do not create a separate topology implementation.

For every theorem-supported quotient already exhausted, construct exactly

P_0(p), P_1(p), P_2(p),
Z_top(p,u),
A_top(p), E_top(p),
source cumulants kappa_1..kappa_4.

Verify independently:

1. partial_s Z_top|_0 = M against the existing matching polynomial;
2. partial_s^2 Z_top|_0 = P_0+P_2;
3. X^3=X reduction of all higher raw source derivatives;
4. full functional duality Z_G(p,s)=Z_Ghat(1-p,-s) on honest tori;
5. the exact even-in-s law on self-matching controls;
6. explicit failure/boundary classification on any degenerate quotients outside the theorem hypotheses.

The source algebra should be a small exact layer on top of existing states, not another exhaustive campaign.

Phase B — topological-source / thermal mixed jet

The two independent coordinates are now source order, not fitted modes:

A_top = <X>,
E_top = <X^2>.

Measure/derive their thermal derivatives jointly:

d_p A_top,
d_p E_top,
d_p^2 A_top,
d_p^2 E_top,
...

with the existing threshold-rank sufficient statistics whenever possible.

Ask whether the anomalous derivative/full-curve state closes in this canonical topology-source jet before adding an arbitrary second scalar bulk mode.

This is one correlated reanalysis when it uses existing histograms, not new evidence.

A sharp outcome is possible:

thermal jet closes in {A_top,E_top}
    -> the extra finite-size state is topology-sector redistribution;

thermal jet needs a third direction even after exact source coordinates
    -> unmarked rank cannot explain it; promote marked/defect/bulk structure.

Phase C — subgroup-valued refinement

Rank is deliberately coarse. The current exact filtration already distinguishes rational winding direction from integral subgroup/saturation data, and #334 proposes marking the first essential birth by its projective line.

Define a richer state sum schematically as

Z_sub(p)
 = sum_{Lambda <= H1(T^2,Z)}
     Prob(im H1 = Lambda) [Lambda],

or an equivalent character/Fourier transform.

Do not assume that nonsaturated integral subgroups are determined by their rational projective line. Store separately:

rank,
primitive rational line/subspace,
integral saturation/index data.

Derive before using it:

• SL(2,Z) / mapping-class action;
• Alexander/Poincare duality action on the subgroup data;
• D4/cover action used by Gaussian tori;
• which contractions reproduce [P2][THEORY/CONTROL] Hexagonal homology-character tomography for H4/H8/H12 #156 primitive-sector characters and [P1 analysis/observable] Projective essential-birth spectroscopy: mark the first ambient-H1 birth by its winding line #334 birth-line marks.

This would turn the line/index marks into coefficients of one representation-valued partition object rather than unrelated observables.

Phase D — relation to surface/cell-complex polynomial literature

There are close precedents but no direct identification should be assumed.

• Krushkal, arXiv:0903.5312, constructs a surface Tutte-type polynomial and a finer version whose coefficients retain subgroups of H_1(Sigma), with surface duality controlled by the intersection pairing.
• Hiraoka--Shirai, arXiv:1602.04561, connect Bernoulli cell-complex persistence, lifetime sums, Tutte polynomials and higher random-cluster models.
• Krushkal--Renardy / Tutte--Krushkal--Renardy cell-complex polynomials (arXiv:1012.1310, 1204.3563) show how Alexander/Poincare duality becomes polynomial duality.
• Duncan--Schweinhart, arXiv:2207.08339 and 2406.08043, use homological fugacities in plaquette random-cluster models.

But #144/PR #229 already established the important obstruction: Matching One is a vertex-subset digital-homology state sum, not an ordinary edge-subset Krushkal/Tutte specialization.

The target is therefore not to undo that negative result. It is to ask whether the local-state/vertex-surface polynomial proposed in #144 admits Z_top or Z_sub as its natural homology-source specialization.

Phase E — compare with Potts-Q tangents only after both are typed

#333 has shown that generic-Q tangents depend on the lift.

The ambient-rank source gives a different, intrinsic derivative direction:

partial_s at fixed Q=1 site ensemble.

Compare it with #258/#262/#263 only after the generic-Q lift/path/projector conventions are explicit.

The useful question is then not

is s secretly Q?

but

which Potts/defect/categorical continuation represents the same topological-source insertion in the continuum, if any?

An obstruction is a valid answer.

Bold conjecture

My strongest current guess is:

Matching One is fundamentally a topological-source response, not a Potts-Q derivative.

The scalar matching observable is the first source moment of the ambient homology image. Potts-Q derivatives are valuable continuum probes, but they are coordinate/lift-dependent ways of moving through a larger theory family. The s source is intrinsic to the Q=1 lattice observable itself.

A stronger version is that the relevant continuum object is a defect/topological-sector chemical potential whose odd source response is restored to zero at the critical fixed point; the observed spin-4 signal is the leading irrelevant violation of that topological-source symmetry.

This stronger statement must earn a defect/CFT realization; it is not part of the exact claim.

Falsification / simplification

• If the {A_top,E_top} thermal jet is algebraically identical to an already-canonical two-coordinate view, record the exact map and use the existing implementation rather than creating a parallel analysis stack.
• If the subgroup refinement adds no information beyond existing primitive-sector descriptors, keep only the rank source.
• If no continuum/topological-defect insertion realizes the s source, retain it as an exact finite/probability generating function; do not relabel it as a local CFT coupling.
• The construction does not imply a closed form for p_c.
• The plaquette random-cluster q fugacity weights internal Betti data; it is a precedent for homological source variables, not automatically the same as this ambient-image-rank source.

Literature anchors

• Arguin, arXiv:hep-th/0111193 — critical torus FK homology sectors and pi_2=pi_0 at Q=1.
• Krushkal, arXiv:0903.5312 — surface polynomial with homology-subgroup refinement and duality.
• Krushkal--Renardy, arXiv:1012.1310; Bajo--Burdick--Chmutov, arXiv:1204.3563 — cell-complex polynomial duality from Alexander/Poincare topology.
• Hiraoka--Shirai, arXiv:1602.04561 — Bernoulli cell complexes, persistent homology, Tutte polynomials and random-cluster models.
• Duncan--Kahle--Schweinhart, arXiv:2011.11903 — ambient giant-cycle homological percolation on tori.
• Duncan--Schweinhart, arXiv:2207.08339; arXiv:2406.08043 — topological random-cluster / duality structures.

Related: #111, #114, #144, #156, #249, #258, #269, #275, #276, #321, #333, #334.

[LightChainr]

Stronger bridge: discrete topological-source values are exact averages over flat cohomology constraints

There is a clean finite-field realization that may turn the auxiliary s source into a genuine twisted/defect-sector object at special source values.

Use mod-q ambient rank, not the integer rank blindly

For a prime power q, define

V_q(omega)
 = im[H1(K(omega); F_q) -> H1(T^2;F_q)],

r_q = dim_Fq V_q in {0,1,2}.

Keep this distinct from the integral/rational rank r. If the integral image is nonsaturated or its index is divisible by q, then r_q can differ from r; that difference is useful information, not a bug.

Let

V_q^ann
 = { alpha in H^1(T^2;F_q) : alpha|_(V_q)=0 }.

Pure finite-dimensional linear algebra gives

|V_q^ann| = q^(2-r_q).

Therefore, configuration by configuration,

q^(1-r_q)
 = (1/q) sum_(alpha in H^1(T^2;F_q))
           1[ alpha|_(V_q)=0 ].

Taking expectation gives the exact identity

E_p[q^(1-r_q)]
 = (1/q) sum_alpha
     Prob_p( every occupied ambient cycle has zero alpha-holonomy ).

The left side is precisely the centered rank-source generating function for r_q at the discrete source value

s = -log q.

So at these points the topological fugacity is an average over flat cohomology/twist constraints.

Why this is useful

This supplies a concrete route for Phase E of this issue:

intrinsic ambient-rank source
    -> finite-field annihilator identity
    -> flat H^1 twist constraints
    -> candidate Potts/loop defect or twisted partition functions.

The first two arrows are exact finite-volume algebra and require no CFT.

For q=2, there are only four cohomology classes on the torus. The three nontrivial classes are explicit parity twists dual to the three nonzero elements of H_1(T^2;F_2). This should be a tiny exact oracle before any continuum interpretation.

Saturation tomography for free

Running the same construction for several small primes is also a direct probe of the integral subgroup data that projective rank loses:

integer rank/subspace fixed,
r_q drops for selected q
    <=> q-primary information is present in the ambient image index/saturation.

Thus the finite-field source naturally refines #334:

projective line        -> rational direction,
{r_q across primes}    -> arithmetic saturation fingerprint,
flat alpha constraints -> twist/character representation.

Do not infer the full Smith data from one prime; derive the exact relation and use enough primes/exponents only when needed.

Literature connection and boundary

Duncan--Schweinhart's plaquette random-cluster / Potts lattice-gauge work (arXiv:2207.08339) gives a rigorous precedent in which homological constraints become Wilson-loop observables; their 2026 arXiv:2607.02434 extends this to topological formulas for Wilson-loop correlations. This does not prove that the square-site alpha|V_q=0 event is already a Potts defect amplitude. It does show that a homology-to-gauge/twist dictionary is mathematically natural enough to pursue.

The immediate task should be exact:

1. extend the digital-Alexander oracle to F_2,F_3 image ranks;
2. enumerate every alpha in H^1(T^2;F_q) and verify the annihilator identity configuration by configuration;
3. compare r_q with the stored integral HNF/saturation data;
4. derive the SL(2,Z) action on alpha and on the constraint vector;
5. only then ask whether the constraint partition functions match known twisted Potts/loop sectors.

If the last step fails, the finite-field identity remains a useful arithmetic/topological decomposition and should not be forced into a CFT defect interpretation.

[LightChainr]

Continuum/TL realization should split into two topological functionals: even noncontractible-loop weight + odd open/closed projector

The intrinsic source has an exact decomposition that I think should guide Phase E before searching for one mysterious seam.

Since

X=r-1 in {-1,0,1},
A_top=<X>=P2-P0,
E_top=<X^2>=P2+P0=1-P1,

we have configurationwise / exactly

exp(s X)
 = 1 + sinh(s) X + [cosh(s)-1] X^2,

and therefore

boxed:
Z_top(s)
 = 1
 + sinh(s) A_top
 + [cosh(s)-1] E_top.

So the full source does not require an infinite new source hierarchy. It requires continuum realizations of exactly two topological functionals.

I think affine Temperley–Lieb literature suggests that these two pieces have different natural realizations.

1. Even source E_top=1-P1: noncontractible-loop / twist trace

In the affine TL/JTL formulation, the zero-through-line standard module carries an independent weight for noncontractible loops, commonly parametrized by

n_nc = z + z^-1.

The torus Potts partition function is reconstructed as a linear combination of traces over standard modules / twists. Rank-one FK topology is exactly the sector in which nontrivial torus winding/interface data are present, so P1 should be reconstructible from a finite combination/projection of these twisted/noncontractible-loop traces.

Do not assume that one derivative with respect to n_nc equals P1: it weights the number/type of noncontractible loops, not merely the indicator of rank one. The exact task is to derive the finite transform

{affine-TL twist / noncontractible-loop traces}
        -> (P0,P1,P2)

using the same homology convention as Arguin/#321.

If such a transform exists, the even source

E_top = 1-P1

has a canonical annular/TL realization.

2. Odd source A_top=P2-P0: cannot come from an ordinary even loop fugacity alone

A standard noncontractible-loop weight does not distinguish exchanging primal and dual 0D <-> 2D; it is an interface/loop functional and is naturally Alexander-even.

But

A_top -> -A_top

under the exact matching/complement duality.

Therefore a single ordinary noncontractible-loop fugacity should not be expected to generate A_top.

Fortunately #321 has already identified the correct odd object:

P_B(Q,p) = Z_2D - Q Z_0D,
P_B(1,p) = A_top.

In the long-cylinder limit this becomes the open-versus-closed periodic-TL sector splitting.

So the natural continuum source pair is

s_even : noncontractible-loop/twist projector -> E_top,
s_odd  : open/closed critical-polynomial projector -> A_top.

The one-parameter intrinsic source s follows the nonlinear curve

coeff_even = cosh(s)-1,
coeff_odd  = sinh(s)

inside this two-dimensional topological source plane.

Exact finite/TL gate

1. In the FK medial-loop representation on the smallest periodic square-bond controls, classify each configuration simultaneously by (r_FK, number/type of noncontractible medial loops).
2. Build the exact polynomial/trace as a function of an independent noncontractible-loop weight z+z^-1.
3. Determine whether P1 is recoverable by a fixed finite linear combination or Fourier/projector transform of twist traces.
4. Prove directly that the ordinary loop-fugacity family is invariant under the 0D <-> 2D dual exchange and therefore cannot generate the odd P2-P0 direction by itself.
5. Add the [P2 theory/Q1↔Q3] Topological phenomenological RG: connect homology balance to periodic-TL eigenvalue crossing and derive the L^-4 shift #321 open/closed projector and verify that the two functionals span {A_top,E_top} exactly.
6. Only then take the continuum limit and identify the corresponding affine-TL modules/defect partition functions.

Why this matters for the current low-rank/Jordan story

This would give the exact two-dimensional rank source a representation-theoretic basis before RG:

A_top : Alexander-odd open/closed splitting,
E_top : Alexander-even winding/interface occupancy.

A 2x2 finite-size transfer on {A_top,E_top} could then be compared with Jordan/ordinary mixing without confusing an arbitrary latent basis with the natural TL topological basis.

It also gives a sharp obstruction: if P1 cannot be recovered from the affine-TL twist/noncontractible-loop trace span, then the intrinsic rank source is strictly finer than the standard annular defect data, and Phase E must use a richer torus/homology defect object.

Literature anchors:

• Böhm–Jacobsen–Jiang–Zhang, arXiv:2202.02986 — affine TL standard modules, noncontractible-loop weights, Markov trace and torus Potts partition function;
• Jacobsen–Ribault–Saleur, arXiv:2208.14298 — twisted torus partition functions and Potts state spaces;
• Jacobsen, arXiv:1507.03027 — open/closed periodic-TL critical-polynomial sector splitting;
• Arguin, hep-th/0111193 — torus homology-sector probabilities.

Bold conjecture: the intrinsic Matching-One source is not one local CFT coupling but a curve in a two-dimensional topological defect-source space. Its odd tangent is the #321 open/closed splitting; its even curvature is a winding/interface fugacity. If true, this supplies a continuum meaning for the exact algebra R[X]/(X^3-X) without invoking a Potts-Q derivative.

[LightChainr]

Overlap resolution with #144: treat the scalar rank-source identification as already achieved there

A concurrent exact result on #144, branch theory/p144-surface-polynomial-classification-20260830, commit 5d8343d, has now established the scalar source more strongly than the opening of this issue:

R_site(a,b;z)
  = sum_B a^|B| b^(N-|B|) z^r_B,

M(a,b)
  = (z d_z - 1) R_site |_(z=1).

Equivalently its Z_rel=sum Q^q with unrestricted q=r_B-1 is exactly the same one-topology-source idea on honest torus carriers. It also proves the stronger local obstruction: ordinary edge/ribbon/transition alphabets cannot represent the four-terminal site state J4 without extra typed site structure, and classifies the result as a site-Krushkal rank-image quotient, not an edge-Krushkal specialization.

So do not develop a second scalar-source implementation here and do not count #337 as an independent discovery/evidence layer for M = source derivative.

Unique scope of #337 after de-duplication

Treat the following as the remaining work:

1. centered source functional and full Alexander functional duality

   X=r-1,
   Z_top(p,s)=E[e^(sX)],
   Z_G(p,s)=Z_Ghat(1-p,-s),
   

   including the complete cumulant-parity hierarchy, rather than only the first derivative at source 1;

2. exact source algebra

   X^3=X,
   exp(sX)=1+sinh(s)X+[cosh(s)-1]X^2,
   

   so the unmarked rank source has exactly the canonical odd/even coordinates (A_top,E_top);

3. finite-field/cohomology-twist realization from the previous comment,

   q^(1-r_q)
    = (1/q) sum_alpha 1[alpha|V_q=0],
   

   which is not contained in the scalar R_site(z) identification and may connect the intrinsic source to twisted/defect sectors;

4. subgroup-valued / saturation refinement, keeping rational line and integral index distinct and asking for SL(2,Z)/Alexander action;

5. thermal-source mixed jet in the canonical (A_top,E_top) plane, as a bounded test of whether the extra finite-size state is already topology-sector redistribution.

The opening conjecture should therefore be read in the narrower form:

#144 supplies the canonical scalar rank-generating polynomial; #337 asks what additional exact structure appears when that source is treated as a full functional, reduced modulo finite fields, or refined from rank to the actual ambient subgroup.

This preserves the useful collision instead of proliferating two names for the same scalar object.

[LightChainr]

Exact result pushed on branch theory/p337-finite-abelian-twist-tomography-20260830, commit 4b17955.

The #269 saturation theorem changes the finite-field route in a useful way. For the global ambient image Lambda and any finite abelian group A of order n,

|{alpha in Hom(Z^2,A): alpha|Lambda=0}| = n^(2-rank Lambda),

hence the full flat-twist trace is

S_n = sum_alpha T_alpha = n^2 P0 + n P1 + P2,
S_n/n = Z_top(s=-log n).

This closes the unmarked source with only two nontrivial source values:

P0=(S3-2 S2+1)/2,
P1=S2-1-3P0,
P2=1-P0-P1.

So order-2 and order-3 twist averages reconstruct the complete intrinsic rank-source functional exactly.

The individual sectors carry the additional information. For nonzero alpha in F_q^2,

T_alpha = P0 + L_ker(alpha),

where L_line is the probability of a rank-one primitive winding line reducing to that projective class mod q. Every projective line has q-1 such twists. Thus the finite-field construction is not Smith-defect tomography: #269 forces r_q=r. It is modular projective-line tomography and gives an exact twist realization of the #334 line mark.

Artifacts include a finite-abelian exhaustive oracle across six group structures, prime projective audits q=2,3,5,7, exact source inversion, line-tomography gates, note, manifest, and scientific card. New tests 5/5 plus #269 regression 5/5 pass. No PR was opened or merged.

Branch: https://github.com/LightChainr/Matching-One/tree/theory/p337-finite-abelian-twist-tomography-20260830
Commit: 4b17955

[LightChainr]

Scientific card — N65 flat-twist sectors from the #334 archive

The exact transform at 4b17955 is now instantiated on production-format data at a7cb19a, branch analysis/p337-p334-flat-twist-n65-20260830 (no PR, no new samples/server).

Sufficiency result. The existing 1714141 tuple (tau1, primitive ell, tau2, DIRECT_RANK2) is complete for every fixed-p F2/F3 constraint sector. tau1/tau2 determine rank 0/2; on the plateau primitive ell mod q determines the rank-one projective class; a direct 0->2 path needs no line. No new field is required.

The scorer reconstructs all 4 F2 and 9 F3 T_alpha sectors for both N65 orientations, plus rank/source coordinates and seven character/projector coordinates, in one 57-dimensional shared-batch covariance block. Every batch/orientation verifies

• T_0=1;
• T_alpha-P0=L_kernel(alpha);
• S_q=sum_alpha T_alpha=q^2 P0+q P1+P2;
• S2/S3 recover P0,P1,P2.

Maximum numerical residual is 2.66e-15.

H4 comparison. The balanced P1(F3) axes-minus-diagonals projector

1/2[(T_axis0+T_axisInf)-(T_diag+ + T_diag-)]

automatically cancels common P0. Its same-modulus orientation contrast is 0.00110298 +/- 0.00150621 (|z|=0.732). Raw phase-aligned chi4 gives |z|=0.729; F2 gives 0.553. F3 is the nominal H4 winner but effectively tied with raw chi4, not a material variance improvement. Its advantage is that it is the canonical minimal balanced flat-twist projector.

New sector. F3 retains a diagonal-odd/reflection-odd character that raw chi4 collapses. Its contrast is 0.00207756 +/- 0.00104063 (|z|=1.996); the joint F3 axis-odd/diagonal-odd diagnostic is 5.903 / 2 df. This is sharper than raw H4 but belongs to a different projective symmetry sector, not a better estimator of the same H4 amplitude.

Lifecycle tuple. observer=flat constraint T_alpha / projective characters; sector=F2,F3; H4-even plus reflection-odd; source=P334 N65 20k; geometry=same-modulus (8,1)/(7,4); dependency=P334_N65_20k_projective_birth_smoke + exact 4b17955; next升权=freeze F3 balanced H4 and diagonal-odd on fresh counters/child size.

Not proved. The character ranking is exploratory reuse, no affine-TL/Potts defect identity or continuum operator/exponent is assigned.

Artifacts: note, score, full covariance. Tests 3/3.

[LightChainr]

Scientific card — exact F2/F3 projective representation decomposition

Branch: theory/p337-flat-twist-representation-20260830
Commit: 3cb06ab

Mechanism space changed:

• On P^1(F2), the D4 image gives 2 A1 + B1, but the realizable S,T image is PSL(2,F2)=S3; the H4 alias and axis-odd coordinates form its irreducible real 2D standard block.
• On P^1(F3), D4 gives 2 A1 + B1 + B2: balanced H4 alias is the second A1 copy, axis odd is reflection-even B1, and diagonal odd is reflection-odd B2.
• With realizable S,T, the three non-scalar F3 coordinates merge into the irreducible 3D standard representation of PSL(2,F3)=A4. Restricting to <T>=C3 gives one neutral line plus a real charged 2D rotation pair. Therefore diagonal odd is one charged-doublet coordinate, not a standalone modular field.

Existing-data placement:

• The a7cb19a vector [H,A,D]=[0.0011029842,0.0051299851,0.0020775613] becomes [u,v,w]=[0.0048254235,-0.0020612155,0.0020775613].
• Full covariance gives charged-pair quadratic 5.9050 / 2 df; neutral z=1.602. The roughly two-sigma D marginal is not promoted to a discovery.

Frozen new discriminator:

• Exact shear transport predicts [H,A,D] -> [0.0021583896,0.0043837008,-0.0028238456] with no fit.
• A paired identity/shear projective-source block should freeze the whole vector and joint covariance. In charged coordinates it must rotate by exp(±2 pi i/3) while the neutral coordinate stays fixed.

Observer/sector/source/geometry: fixed-p N65 projective rank-birth flat-twist vector; same-modulus Gaussian orientation pair; dependency group P334_N65_20k_projective_birth_smoke.

Does not prove: continuum field identity, independent replication, or a new effect.

Artifacts: notes/p337-flat-twist-representation.md, exact machine certificate in results/local-20260830/P334-projective-birth-N65-smoke/flat_twist_representation.json, and 4 focused tests.

[LightChainr]

Scientific card — the frozen F3 shear matrix is now an end-to-end exact gate

Branch/commit: analysis/p337-shear-transport-smoke-20260830 / bd13913.

Archive audit: no pre-existing legal identity/T-shear projective-birth pair exists. The P334 archive contains two physical Gaussian orientations, not two period bases of one shape.

I therefore ran a fresh local N65 smoke after the matrix was frozen at 3cb06ab: 10k counters, 20 aligned batches, both (8,1) and (7,4), with P_shear=P_identity*T^-1. No Huawei resource was used.

Exact result:

• every sparse (tau1,tau2,kind,ell,count) cell obeys ell_shear=primitive(T ell_identity) in all 40 shape/batch blocks;
• fresh identity contrast [H,A,D]=[-0.0056598827,-0.0004602931,-0.0002831596];
• frozen matrix predicts [-0.0001252523,-0.0043738678,-0.0036304151];
• directly reconstructed shear archive gives the same vector;
• mean 3D residual [4.88e-19,-8.79e-18,-7.98e-18], maximum batch residual 6.55e-17.

The artifact retains full 6x6 identity/shear covariance and 3x3 residual covariance. No chi-square is quoted because the residual is pathwise exact and its covariance has rank zero apart from floating roundoff.

Mechanism-space consequence: right period-basis shear is a relabeling of the same quotient graph. This closes the convention chain but is not independent charged-sector evidence. Larger production on this pair has zero information value. A nontrivial next test needs an explicit finite-field twist/defect source whose sheared version is not a deterministic line relabeling.

Lifecycle tuple: observer=F3 projective triplet; sector=A4 standard / T charged split; source=unweighted P334 birth stream; geometry=N65 identity versus right-basis shear; dependency=fresh P337 10k synchronized counters; next升权=physical twist/defect source insertion.

Artifacts: notes/p337-flat-twist-shear-transport-smoke.md, results/local-20260830/P337-flat-twist-shear-smoke/score.json, scorer and 2 focused tests. Representation and prior flat-twist regressions also pass.

[LightChainr]

Scientific card — complete p-dependent F3 constraint partition curves

Branch/commit: analysis/p337-f3-twist-curves-20260830 / 62b2b86.

No new production was required. The P334 tuple (tau1,ell,tau2,DIRECT_RANK2) determines the full degree-65 Bernstein curve for every F3 twist:

• T_00(p)=1 exactly;
• T_alpha(p)=P0(p)+L_kernel(alpha)(p) for nonzero alpha;
• analytic derivative N sum_k [c(k+1)-c(k)] B_(N-1,k)(p);
• eight nonzero twists retained, with four identifiable kernel curves because alpha and 2 alpha are proportional.

The artifact stores aligned per-batch coefficients sufficient for arbitrary-p and cross-p covariance, plus a central grid with full 48x48 covariance across both physical orientations, all twists, derivatives and [H,A,D] characters.

Exact transport:

• every nonzero twist is monotone and zero twist is constant;
• complement maps P0<->P2, keeps L_line even under p->1-p, hence C_dual(p)=C(1-p) and C_dual_prime(p)=-C_prime(1-p);
• D4 acts as S:(H,A,D)->(H,-A,-D) and reflection (H,A,D)->(H,A,-D).
  Maximum gate residual is 2.50e-15.

Exploratory geometry selector:

• the only central character crossing with 20/20 leave-one-batch bracket survival is H_(7,4)(p)=H_(8,1)(p);
• p_cross=0.573633326, slope 0.0602462, jackknife SE 0.02281;
• exact complement partner p_cross_dual=0.426366674, slope -0.0602462.

The root is only 0.01911 from p_ref, less than one current SE. Inverse-sample scaling gives about 114k samples/shape for 2 sigma and 256k for 3 sigma. These are design numbers only; no sampling was added.

Mechanism changed: the explicit twist observable is now a complete curve, not a one-p diagonal-odd score. The H crossing is the unique frozen geometry target from this archive. It is not a resolved root split, universal crossing or independent defect response.

Artifacts: notes/p337-f3-twist-curves.md, results/local-20260830/P337-F3-twist-curves/score.json, curve scorer and 2 focused tests.

[LightChainr]

Scientific card — exact N13 bridge from F3 twists to primitive A4 current

Branch/commit: theory/p337-p334-n13-twist-flux-bridge-20260830 / 770fe27.

On the exact 3+2i N13 quotient, choose one nonzero F3 twist per projective kernel. Coefficient by coefficient in the degree-13 Bernstein curve,

H_F3=T_01+T_10-T_12-T_11=L_axis_x+L_axis_y-L_diag_plus-L_diag_minus.

The physical line characters are fixed:

z_axis=(-119+120 i)/169, z_diagonal=-z_axis.

Therefore the complete polynomial identity is

A4(p)=z_axis H_F3(p).

The unit-vector convention stored by a7cb19a is H_unit=H_F3/2, so its dictionary is A4=2 z_axis H_unit. The factor two is normalization, not a fitted coupling.

The state-current bridge also closes at every lower size k:

• dH_F3/dp=J_H,birth1-J_H,exit2;
• J4_birth1=z_axis J_H,birth1;
• J4_exit2=z_axis J_H,exit2;
• dA4/dp=z_axis dH_F3/dp.

All state and source/sink identities pass coefficientwise. Thus on this quotient the F3 H derivative and P334 primitive birth/exit current are the same current in two exact bases, not separate mechanisms.

New selection rule: both A=L_axis_x-L_axis_y and D=L_diag_plus-L_diag_minus vanish separately at every state, birth and exit coefficient. This extends algebraically to every unweighted Gaussian-ideal quotient because P(a,b)=[[a,-b],[b,a]] commutes with quarter-turn R, while A and D are R-odd.

Consequently the approximately two-sigma N65 diagonal-odd marginal in a7cb19a is now assigned to an exact symmetry-null direction for the unweighted ensemble, not retained as a candidate physical response. A nonzero A/D test requires an explicit R-odd charged insertion or a non-Gaussian quotient without this automorphism.

Does not prove: continuum field identity or H scaling exponent.

Artifacts: exact certificate, note, machine science card and 4 focused tests; upstream N13 flux tests 5/5 also pass.

[LightChainr]

Exact continuation of 770fe27: commit 539b629 constructs the minimal explicit R-odd F3 defect on Gaussian N13.

Representation: on P1(F3), physical quarter-turn acts as (axis-x axis-y)(diag+ diag-). The two missing charged directions are q_A=T_01-T_10 and q_D=T_12-T_11. Both have R eigenvalue -1, hence projective C4 charge 2; under D4 they are B1 and B2. They are two copies of the odd one-dimensional block, not a +/-i doublet.

Insert u^q_C with u=omega, omega^3=1. Then O_C(omega)=E[q_C omega^q_C]=(omega-omega^2)W_C/2, W_C=E[q_C^2]. This activates A or D while preserving exact selection rules: an A source has only A response, a D source only D response, H and A-D cross-response remain zero.

At p_ref the exact integer-charge responses are W_A=0.319893954769 and W_D=0.0201214105295. The normalized H/A/D response matrix is diag(0.159946977384, 0.0100607052647) in the A,D block, zero elsewhere. The former unweighted D null is not promoted; this is a different, explicitly charged observable.

The exact oracle gives every Bernstein state coefficient and birth/exit response, with dW_C/dp=J_C,birth-J_C,exit coefficientwise. Minimal next production tuple: (W_A,J_A,birth,J_A,exit,W_D,J_D,birth,J_D,exit), one covariance block. Existing projective-line archives suffice; no new path field is needed.

Boundary: exact finite charged-defect response and selection rule, not yet a continuum operator identity or scaling claim. Science card and machine certificate are in the commit. No large computation or Huawei machine was used.

[LightChainr]

Exact #334→#337 bridge at 2ec9f19: finite-character geometry and projective timing factorize.

For real orbit-net currents J1,J2 and fixed complex characters chi1,chi2,

Re[(chi1 J1) conjugate(chi2 J2)] = Re[chi1 conjugate(chi2)] J1 J2 = Gram(chi1,chi2) J1 J2.

The N7 asymmetric HNF exact falsifier retains the close paired timing zeros but flips the alignment window relative to Gaussian N13/N17 because its Gram is +527/625 rather than negative. In #337 language, the finite-field/ambient character block supplies the Gram factor; #334 source/sink dynamics supplies J1J2. A twist-character change can therefore flip reinforcement without changing the timing-zero mechanism. No production or continuum identification is involved.

[LightChainr]

The frozen charged sources from 539b629 have now been revealed from the existing N65 archive, with no new simulation and no reselected contrast: commit 83207c7.

At p_ref, both explicit source susceptibilities are easy to resolve:

• 8+i: W_A=0.3362043 +/- 0.002048, W_D=0.0370407 +/- 0.001040.
• 7+4i: W_A=0.3383445 +/- 0.001897, W_D=0.0369750 +/- 0.000645.

The F3 phase remains parameter-free: O_C(omega)=(omega-omega^2)W_C/2, so both one-points lie on the same +i ray. q_A/q_D retain projective C4 charge 2 and D4 B1/B2. Direct T permutation of the four line bins matches the frozen H/A/D shear matrix for both state and response matrices to 4.2e-17.

Null controls are as predicted: A-D cross response is zero statewise; H charged cross-response reduces to the sampled unweighted R-odd one-point. The latter joint null quadratics are 4.510/2 and 3.207/2 for the two shapes. Current continuity holds to 4.1e-15.

The scientific split is useful: charged activation is precise, but same-N orientation modulation lives mainly in A source timing, not its plateau alone. Frozen within-batch scores are A (W,birth,exit)=12.153/3 df, D=1.509/3 df, joint six-vector=15.530/6 df; W_A alone is only 0.900 sigma. This is an engineering-block reveal, not a continuum or independent shear claim. Full batch covariance and values are in the certificate.

[LightChainr]

The #334/#337 factorization is now atlas-level at 62edfe7, not a one-quotient observation.

Across all six HNF index<=12 quotients passing the flux-blind two-orbit gate, line-orbit compression and the full polynomial identity

Re[(chi1 J1) conjugate(chi2 J2)] = Gram(chi1,chi2) J1 J2

hold coefficientwise. Four positive-Gram N7 HNFs have reinforce/cancel/reinforce topology; two negative-Gram N8 HNFs have the exact reverse. The projective timing layer supplies J1J2; the finite-character/ambient layer supplies Gram. Root closeness itself fails at N8, so it must not be folded into the representation statement.

Scientific boundary: exhaustive only under the declared HNF gate through N12; two exact signatures after symmetry copies; no production or continuum identity.

[LightChainr]

The same 1714141 archive has now been decomposed in the frozen orthogonal current basis (W, J_minus=birth-exit, J_plus=birth+exit), with full batch covariance and no new simulation: f3384f4.

For the N65 second-minus-first contrast:

• A: W=0.0021403 +/- 0.0023787 (0.900 sigma), J_minus=0.0970158 +/- 0.0286164 (3.390 sigma), J_plus=0.0268309 +/- 0.0501771 (0.535 sigma).
• D: W=-0.0000657 +/- 0.0013698, J_minus=-0.0105921 +/- 0.0090356, J_plus=-0.0128512 +/- 0.0379713.

The invertible transform preserves the A full score at 12.153/3 df. Net alone carries 11.494; common activity alone 0.286. Conditioned on (W,net), activity adds only 0.236. D remains 1.509/3 df. Thus the charged orientation response is specifically A net timing, not a common-activity amplitude.

Standardized covariance eigenmodes agree: the two A modes carrying large net components contribute 5.477 and 6.048, while the activity-dominated mode contributes only 0.628. D has no scored eigenmode.

#334 crosswalk is structural only: J_plus is common activity and J_minus the source-sink counterflow residual. A/B1 is intra-axis, unlike the inter-orbit H4 coordinate. The 381984d line-sorting score reuses this exact N65 block; 77aa3fe exact N13/N17 is a separate dependency group. No quadratics or significance are pooled.

[LightChainr]

The timing factor in the character-Gram factorization now has an exact root criterion.

Commit be32ae0 proves that one sign change in the normalized rank-one state-count slope q_{k+1}-q_k forces one transverse projective-timing zero. Across every connected honest-face HNF quotient through N=12, all 240 line sequences and 217 orbit sequences satisfy strict ULC and decreasing adjacent ratios, but that regularity remains finite exact evidence/conjecture rather than a general theorem.

This cleanly separates the two factors already visible in #337:

1. ambient/finite-field character geometry fixes the Gram sign;
2. projective timing fixes the sign of J_1J_2 and, under the exact sign-change criterion, contributes one simple zero per orbit.

A literal site-matroid proof is impossible: diag(2,2) has a one-site homology-rank jump 0->2. Difference-of-increasing-events is exact but insufficient for LC, so the next theoretical target is the stronger order-convex normalized-matching/Lorentzian structure.