analysis: integrate topological scaling program and solvable rank-source sector

[LightChainr]

Summary

Owner-delegated integration of the 2026-09-14 analysis rounds. This remains a draft analysis PR: no merge requested, no STATUS claim upgrade, no original-U identification.

The branch now turns several previously separate threads into one typed program:

geometry / balance root
-> complete winding components
-> topological pivotal geometry
-> rank-simplex / source calculus
-> process-level two-birth persistence
-> critical modular homology controls
-> matching-odd continuum selection questions.

analysis/topological-scaling-program-20260914.json is the current machine-readable index.

Exact finite algebra now collected here

For X=rank-1 in {-1,0,1}:

1_{r=2}=(X^2+X)/2,
1_{r=1}=1-X^2,
1_{r=0}=(X^2-X)/2.

Canonical three-state coordinates:

b = 1/2 log(P0/P2),
d = log[P1/sqrt(P0P2)],
P = (e^b,e^d,e^-b)/(2 cosh b+e^d).

A bounded topological source sX acts exactly as b->b-s, leaving d invariant. At balance the odd/even rank directions are Fisher-orthogonal and span the whole nontrivial rank sigma-algebra.

For any exponential physical source A, after profiling the thermal/root nuisance,

beta_A = Delta E[A]/Delta E[K] = E[X A]/E[X K],
R_A = A-beta_A K,
z_*'(0) = -beta_A,
z_*''(0) = -Delta Var(R_A)/Delta E[K].

This supplies an exact finite source quotient by span{1,K}.

Information geometry and process layer

The rank simplex with Fisher metric is the positive octant of a radius-2 sphere. The branch now contains:

• intrinsic geodesic curvature of the physical rank-law curve;
• total Fisher length and Fisher-sphere area;
• exact rank ANOVA odd/even decomposition;
• Fisher-area/canonical-persistence bound E G >= A_F^2/8;
• monotone-likelihood-ratio order K|rank2 >=_MLR K|rank0, with g1=W1(mu2,mu0);
• a two-birth persistence representation (T1,T2) / (J1,J2);
• exact L=3 all-permutation copula controls and sharp marginal-only transport bounds;
• exact one-step flux decomposition where the only process quantity hidden from static marginals is direct 0->2 flux B_k.

The canonical process conjecture is that (B1,B2)=(b(T1),b(T2)) has a lattice-independent fixed-modulus limit; raw insertion gaps then naturally scale as L^(5/4).

Critical continuum controls

The branch includes direct Arguin/Pinson Q=1 torus homology sums for arbitrary modulus, not just the square torus. At criticality P0=P2 for every modulus. For rectangular tau=i r, the bounded topological-source zeros collide at

r_* = 1.7878293526796570376290825...

(and its reciprocal), where (P0,P1,P2)=(1/4,1/2,1/4). This is a static closure-polynomial exceptional point, explicitly not a Jordan diagnostic.

The generic-Q extension gives the exact topological chemical-potential shift

P2=Q P0,
b_c(Q)=-1/2 log Q,

plus a source-invariant even rank-shape response. Q=4 is treated as the real Coulomb-gas endpoint and is not symmetrically differentiated.

A further modular conjecture is now delegated to #781: for Q=1, c_*(tau)=P1/(2P0) appears to attain its global minimum at the hexagonal modulus.

Strong conjecture and its explicit threat

The current high-risk hypothesis is that the first nonzero matching-odd scalar correction is the spinless 8-arm / 8-leg field x=21/4, explaining the L^-4,L^-6,... root-correction tower.

This is not promoted to a result. Exact L=3,4 pivot controls find abundant rank-jump-two theta/T3 spines; at L=4 they dominate the jump-two class. These are genuine 6-arm candidates. Therefore the 8-arm hypothesis can survive only if #768 finds a real matching/map-sector cancellation of the lower 6-arm channel.

Active delegated tasks

• [P0｜深度理论/检索] 首个非公共扇区修正：标量8-arm、热族spin4与低阶通道的实际矩阵元 #768 — matching-odd sector audit and 6-arm kill test.
• [P1｜当前L5封口] pivotal-pair原始联合表；不追加L6或新描述符 #769 — exact pivotal-pair atlas / cardinality-resolved direct-jump flux / L5 controls.
• [理论/有界计算] 独立热正规坐标与 Harris 正控制：用质量 κ 校准 M''/(2M') #770 — independent mass-defined thermal coordinate and Harris controls.
• [有界计算] complete-component 质量–边界锁定：K/B 联合 Hessian、Palm score 方差与近临界 composition ray #774 — complete-component K/B covariance, Palm Ward locking and composition CLT diagnostics.
• [P1｜完成当前最小产物后停] rank-sector原始系数：取消L5–8自动阶梯 #775 — exact rank-sector occupation polynomials L=5–8 and all derived Fisher/process checks.
• [严格正控制] self-matching triangular-site：有限 rank-source 对称、near-critical 方程与 square-site universality 对照 #776 — independent triangular self-matching positive control and near-critical torus interface.
• [现有数据/过程级] two-birth persistence copula：恢复 (J1,J2)、canonical gap 与 simultaneous-rank birth 原子 #778 — existing-data paired two-birth copula; transport-bound fallback if paired raw is absent.
• [理论/模函数] critical torus rank-law 极值：hexagonal modulus 是否全局最小化 c*(tau) 与 source-zero 距离 #781 — modular/theta proof or counterexample for the hexagonal rank-law extremum.

Boundaries

• Base remains main@d31fa5fe; unmerged Geometric balance manuscript: sharp full-law criterion for arbitrary integer periods #739/Multi-advance 20260913: spatial-source Hessian, invisible marks, annealed/quenched roots, rank-source zero-free strip, conditional-odds integration, limit-order audit (do not merge) #737/theory(p586): graded Q->1 tangent benchmark (B_even vs ambient homology) #746 are cited by pinned commit and are not silently integrated.
• No new Monte Carlo, GPU, numerical pc claim, STATUS edit or original-U identification.
• Square-lattice arm exponents remain a universality input where used.
• Tiny L=3,4 square/triangular agreement is a positive control, not an asymptotic proof.
• One early triangular result JSON had manually mistranscribed higher cumulants; the script was correct and the JSON was corrected in a later commit. Git history preserves the mistake. A local Q=4 symmetric derivative attempt was also rejected and the committed result marks that endpoint as not differentiated.

Full repository CI has not been run for this additive analysis branch.

[LightChainr]

Post-open extension committed on the same branch: added exact source-quotient geometry and extended topological_rank_source_control.py/JSON with conditional K cumulants in rank-0/rank-2 and the sourced-root logit derivatives. The new exact generic source formula at a balance root is dz_*/dη = -ΔE[A]/ΔE[K] = -E[XA]/E[XK] for an exponential source score A; the PR note records the special mean-zero spatial case z_*''(h)=-E[XH^2]/E[XK]. Also added the finite nuisance quotient projecting a candidate score onto span{K-EK, X} before any continuum naming. Manifest hashes were refreshed at commit 3c3ca78964db3528cfbd01a654aa60223c0702ac. No claim status changed.

[LightChainr]

Round-6 direct integration update complete on this draft branch. New exact/analysis files after PR open:

• notes/topological-scaling-program-20260914-addendum.md: general balance-line source calculus, Fisher rho_KX^2~L^-1/2 conditional scaling, near-critical two-variable rank-source equation, derivative parity ladder, conditional-density CLT.
• notes/rank-simplex-canonical-coordinates-20260914.md: global exact (b,c) coordinates, source action b->b-s, c invariant.
• notes/rank-simplex-self-normalized-curve-20260914.md: parameterization-free c=C_L(b), exact intrinsic curvature, root-vs-c-slope exponent relation.
• notes/rank-jump-two-attachment-spines-20260914.md and ...arm-extraction...: exact spine classification + covering-space black-arm proof draft.
• exact controls topological_rank_source_scaling_checks.py/json and rank_jump_two_spine_control.py/json.

The most material correction is now explicit: L4 exact single-site rank-jump-two has 289 configs, with T3 theta 176 / T4+ 92 / split 21; at p=0.59274605079210, T3 carries 60.07% of jump-two probability. Therefore rank jump two => four black arms => 8-arm is false. The 8-arm explanation can survive only through a genuine matching-odd map/parity/fusion cancellation of the abundant theta/6-arm channel; #768/#769 were updated accordingly.

New delegated bounded tasks: #774 K/B Palm composition locking, #775 exact rank-sector polynomials L5–8, #776 triangular self-matching rigorous positive control. Manifest updated at e433e20f. No STATUS / pc / original-U claim upgrade.