theory(p586): graded Q->1 tangent benchmark (B_even vs ambient homology)

[LightChainr]

Summary

Positive-control-first graded Q->1 tangent benchmark for #586. Reuses the exact #581 finite-volume gate (enumerate_torus + covariance_split) for the lattice control, and adds the Phase A continuum internal check (differentiate the BPZ/(2,1) ODE, verify the Q-tangent satisfies the inhomogeneous ODE).

Key numbers

• Lattice graded split exact: Cov(O,T) = 1/2 Cov(O,B_even) + 1/2 Cov(O,X) with split_is_exact=True for all six observables at L=2 (256 configs) and L=3 (262144 configs). Identity gate T - T* = X has 0 failures.
• open_edges loads purely on X (B_even = 0); wrap_either/wrap_cross carry opposite-sign B_even pieces (∓427/65536) with identical X -> B_even and X are physically distinct, not aliased.
• Phase A continuum internal check passes: Q-tangent of a representative (2,1)-degenerate BPZ ODE satisfies the differentiated inhomogeneous ODE, residual max|R|/||G|| = 4.4e-6 (threshold 1e-3).
• Phase B bulk-logarithmic numeric declared a buy-back (VJS / Camia-Feng generic-Q data not in tree). Spin-4 tangent reopening of [P2 analysis] Linearized Potts crossing at Q=1: reconstruct the logarithmic theory as a tangent CFT #263 not yet earned.

Two measure-score contributions reported separately

The B_even and X contributions are reported separately and only summed afterwards, never recombined before reporting.

Decision

Promote the graded tangent machinery as an exact method-level diagnostic for #581 scale tomography. Narrow spin-4 tangent reopening of #263 remains conditional on the Phase B bulk-log positive control, a declared buy-back.

Deliverables

• notes/p586-graded-q-tangent-20260913.md
• scripts/graded_q_tangent_benchmark.py
• results/graded-q-tangent-20260913/ (REPORT.md, metadata.json, commands.txt, raw/, derived/)

Full Matching-One repository CI has not been run for this commit.

[LightChainr]

New continuum generic-Q homology control committed on draft #773: notes/generic-q-topological-source-20260914.md. Arguin's critical FK homology relation gives P2=Q P0 exactly, hence canonical odd coordinate b=1/2 log(P0/P2)=-1/2 log Q for every torus modulus. A topological source sX translates b->b-s, so s_Q=-1/2 log Q exactly rebalances rank0/rank2. The source-invariant even coordinate d=log[P1/sqrt(P0P2)] carries the nontrivial Q response. At Q=1,tau=i, dd/d log Q=-0.209344181878...; in the sourced-balanced Fisher metric the continuum rank-law Q tangent is ~93.74% odd / 6.26% even by squared norm. Suggested graded-Q check: push the lattice tangent to (P0,P1,P2), transform to (b,d), require odd derivative -> -1/2 before interpreting the residual even piece. Q=4 is an endpoint and is explicitly not symmetrically differentiated.

[LightChainr]

Exact identifiability boundary added on draft #773: after any compensating source/root adjustment that restores P0=P2, the aggregate three-state rank law lies on the one-dimensional Fisher geodesic B={(a,1-2a,a)}. Hence every balanced even perturbation has the same aggregate-rank tangent (+1,-2,+1), and the full unparameterized rebalanced path is only a reparameterization of this same curve. So dd/dlogQ is a valuable normalization/positive-control scalar but cannot by itself identify B_even with a unique continuum field; another marked/map/spatial observable is required. See notes/balanced-rank-identifiability-no-go-20260914.md.

[LightChainr]

generic-Q tangent gets canonical coordinates: eta=h+1/2 log Q

A new exact algebraic reparameterization is on #771: canonical-potts-topological-coordinates-20260914.md (commit 523354a20f9469b9e0b744f7a4e1213681a614f4).

Using the torus Euler identity, define

x   = v/sqrt(Q),
eta = h + 1/2 log Q.

Then, up to a global factor, the sourced FK weight is exactly

W(A) ∝ (sqrt Q)^{b(A)+1_(r!=1)} x^{|A|} e^{eta(r-1)}.

So the three coordinates separate cleanly:

Q       = common loop fugacity,
log x   = thermal/self-duality coordinate,
eta     = primal/dual topological charge.

The graph-polynomial source h_Q=-1/2 logQ is simply eta=0. Therefore your exact kinematic dh/dlogQ=-1/2 term has a more structural meaning: it is precisely the compensating source motion required to remain on the primal-dual symmetric eta=0 surface while Q changes.

After subtracting that piece, the Q tangent really probes common loop fugacity / local coupling / module content rather than a trivial topological imbalance. On a self-dual lattice, duality becomes (log x,eta)->(-log x,-eta) up to the usual global factor.

This seems like the cleanest coordinate system for the graded tangent programme; I would avoid raw (v,h) derivatives once generic Q is involved.