diff --git a/docs/manuscripts/geometric-balance/branch-endpoint-dressing-beta-law-20260914.md b/docs/manuscripts/geometric-balance/branch-endpoint-dressing-beta-law-20260914.md
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+# Branch endpoint dressing, singularity order, and the split-fraction Beta law
+
+Date: 2026-09-14
+
+Status: exact convolution/Tauberian algebra plus a mechanism diagnostic for #758/#762/#800. It generalizes the equal-rate exponential/Uniform-split picture in `span-double-pole-branch-convolution-20260914.md`.
+
+## 1. One-sided branch ansatz
+
+Suppose a one-sided excess branch of integer length h has asymptotic weight
+
+```text
+b_h
+ ~ C/Gamma(alpha) * h^(alpha-1) rho^h,                   (1.1)
+```
+
+with `alpha>0` and `0<rho<1`.
+
+Equivalently, under the usual transfer/Tauberian regularity, its generating function has singular part
+
+```text
+B(z) ~ C (1-rho z)^(-alpha).                             (1.2)
+```
+
+The exponent `alpha` packages endpoint dressing/rooting information in the longitudinal variable. It is not asserted here to be a universal exponent of the SITE model.
+
+## 2. Two-ended convolution
+
+For two asymptotically independent ends with the same `alpha,rho`, the total excess weight is the convolution
+
+```text
+d_h = sum_{k=0}^h b_k b_(h-k).
+```
+
+Therefore
+
+```text
+B(z)^2
+ ~ C^2 (1-rho z)^(-2 alpha),                             (2.1)
+```
+
+and
+
+```text
+d_h
+ ~ C^2/Gamma(2 alpha) * h^(2 alpha-1) rho^h.             (2.2)
+```
+
+Thus the polynomial/span prefactor and the one-sided endpoint exponent are the same datum in two languages.
+
+Special cases:
+
+```text
+alpha=1/2 : d_h ~ const * rho^h          (simple-pole order),
+alpha=1   : d_h ~ const * h rho^h        (double pole / Erlang-2),
+alpha=3/2 : d_h ~ const * h^2 rho^h      (third-order singularity).
+```
+
+The effective near-double-pole pattern in #800 therefore points naturally toward `alpha` near one, subject to direct operator confirmation.
+
+## 3. Conditional branch split
+
+Let
+
+```text
+U_h = A_-/(A_-+A_+)
+```
+
+conditional on total excess `A_-+A_+=h`.
+
+Using (1.1), for `k=uh` away from the endpoints,
+
+```text
+P(A_-=k | A_-+A_+=h)
+ proportional to
+ k^(alpha-1)(h-k)^(alpha-1).
+```
+
+The Riemann-sum limit is
+
+```text
+boxed:
+U_h => Beta(alpha,alpha),                                 (3.1)
+```
+
+with density
+
+```text
+u^(alpha-1)(1-u)^(alpha-1)/B(alpha,alpha).
+```
+
+Therefore the morphology and the singularity order cross-check one another:
+
+```text
+span prefactor h^(2alpha-1)
+<=> generating singularity order 2alpha
+<=> branch split Beta(alpha,alpha).                       (3.2)
+```
+
+## 4. Why alpha=1 is plausible for a free branch endpoint
+
+A fixed-root to **fixed-point** two-dimensional Ornstein--Zernike connection typically has a longitudinal `h^-1/2` prefactor, suggestive of `alpha=1/2` if that fixed endpoint were the branch observable.
+
+A complete-component branch endpoint is not pinned to one transverse site. In the simultaneous regime where the transverse Gaussian spread is `O(sqrt(h))` and is much smaller than the circumference, summing over the `O(sqrt(h))` typical endpoint locations cancels the point-to-point `h^-1/2` local-CLT factor. This heuristically restores
+
+```text
+b_h ~ const * rho^h,
+```
+
+that is `alpha=1`.
+
+This is a mechanism argument, not a proof for square SITE. Near criticality, endpoint arm insertions can also change amplitudes in the correlation length; those must be separated from the power of h.
+
+## 5. A three-way falsification test
+
+The same `alpha` can be estimated three ways without fitting an arbitrary morphology model:
+
+### Spectrum
+
+Fit the late span response to
+
+```text
+h^(2alpha-1) rho^h
+```
+
+or the corresponding singularity order.
+
+### Hazard
+
+From (2.2),
+
+```text
+gamma_eff(h)
+ = gamma - (2alpha-1)/h + O(h^-2).                       (5.1)
+```
+
+### Morphology
+
+At fixed large excess-span bins, fit/test
+
+```text
+U = A_-/(A_-+A_+) ~ Beta(alpha,alpha).                    (5.2)
+```
+
+Agreement of the same alpha across (5.1)--(5.2) would be a substantially stronger mechanism test than a single span exponent.
+
+In particular:
+
+- `alpha=1`: Uniform U and double-pole/Erlang span;
+- `alpha=1/2`: arcsine U, endpoint concentration, and simple-pole total span.
+
+This makes the planned #762 U statistic directly informative about endpoint sewing/prefactor structure relevant to #740/#758.
+
+## 6. Claim boundary
+
+- Exact algebra under ansatz (1.1): Sections 2--3 and hazard expansion (5.1).
+- Mechanism hypothesis: free transverse endpoint summation yields `alpha=1` for the long SITE branch.
+- Not claimed: the one-sided branch factorization, the value of alpha for actual SITE complete components, or equality with a fixed-end two-point OZ amplitude.
diff --git a/docs/manuscripts/geometric-balance/span-double-pole-branch-convolution-20260914.md b/docs/manuscripts/geometric-balance/span-double-pole-branch-convolution-20260914.md
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+# A two-ended branch explanation for the span doublet
+
+Date: 2026-09-14
+
+Status: a mechanism-level conjecture motivated by the visible-pole diagnostic in `span-visible-pole-splitting-20260914.md` and by the loop--branch decomposition in #758. It is more specific than the generic metastable-doublet interpretation and yields direct tests for #762.
+
+## 1. The algebraic signature
+
+The stored complete-component span spectra show two slow positive visible poles
+
+```text
+rho_+ > rho_-
+```
+
+with nearly opposite scalar residues. This is exactly the signature produced by a convolution of two one-sided geometric/exponential waiting laws.
+
+If two nonnegative branch lengths `A_-`, `A_+` have generating functions
+
+```text
+B_-(z) ~= C_-/(1-rho_- z),
+B_+(z) ~= C_+/(1-rho_+ z),
+```
+
+then their sum has
+
+```text
+B_-(z) B_+(z)
+ ~= const / [(1-rho_- z)(1-rho_+ z)].                    (1.1)
+```
+
+The coefficient is a difference of two exponentials with opposite residues:
+
+```text
+[z^h] B_- B_+
+ proportional to
+ (rho_+^(h+1)-rho_-^(h+1))/(rho_+-rho_-).                 (1.2)
+```
+
+As the two one-sided masses coalesce, `rho_+-rho_- -> 0`, (1.2) tends to
+
+```text
+const * (h+1) rho^h.                                      (1.3)
+```
+
+Thus a near-double pole naturally produces both observations in #800:
+
+- coalescing visible roots;
+- residues with ratio near `-1`.
+
+No metastable tunnelling interpretation is required for this algebra.
+
+## 2. Geometric interpretation for complete winding components
+
+In the saturated large-span branch of the #758 candidate, write schematically
+
+```text
+L = L_core + A_- + A_+,
+```
+
+where `L_core` is the vertically saturated winding core and `A_-`, `A_+` are lower/upper longitudinal decorations or branches.
+
+The rate candidate above saturation pays a linear cost only for the **total** excess branch length. This leaves a one-dimensional split degeneracy between the two ends. If the two far branch excursions are asymptotically separated by the saturated core and share the same one-sided cylinder mass, then the natural first approximation is precisely the convolution in Section 1.
+
+This gives a direct bridge:
+
+```text
+#758 linear branch rate
+        +
+two-ended branch split
+        ->
+double-pole / Erlang-type subexponential factor.          (2.1)
+```
+
+The LDP exponent can therefore be correct even when the fixed-w scalar response never looks like a single exponential on height `O(w)`.
+
+## 3. New subexponential prediction
+
+In the exactly coalesced idealization,
+
+```text
+d_h ~= C h e^{-gamma h}.                                  (3.1)
+```
+
+Consequently the tail also has a linear polynomial factor,
+
+```text
+P(L>=h) ~= C' (h+c0) e^{-gamma h},                        (3.2)
+```
+
+and the finite-height hazard satisfies
+
+```text
+gamma_eff(h)
+ = -log[T(h+1)/T(h)]
+ = gamma - 1/h + O(h^-2)                                  (3.3)
+```
+
+in the pre-splitting regime `h |gamma_+-gamma_-| << 1`.
+
+For `h=A w`, the `log h` prefactor contributes only `O(log w/w)` to the finite-size rate and therefore does not alter the #758 linear LDP slope. It does, however, matter for prefactor diagnostics and for any attempt to identify the ultimate fixed-w pole from moderate heights.
+
+The current NN `p=1/4` hazards are qualitatively consistent with this correction; this note does not claim that the coefficient `1` in (3.3) has already been numerically certified for the full SITE model.
+
+## 4. A direct #762 prediction: uniform branch split
+
+If in the asymptotic branch regime the two one-sided excesses are independent exponentials with the same rate,
+
+```text
+A_- ~ Exp(gamma),
+A_+ ~ Exp(gamma),
+```
+
+then conditional on their total
+
+```text
+R=A_-+A_+,
+```
+
+the fraction
+
+```text
+U=A_-/(A_-+A_+)
+```
+
+is exactly
+
+```text
+boxed: U | R  ~ Uniform(0,1).                              (4.1)
+```
+
+Equivalently `(A_-/R,A_+/R)` is `Dirichlet(1,1)`.
+
+This turns one of #762's morphology options into a sharply motivated test rather than a generic candidate. In the saturated branch regime, the two-ended convolution mechanism predicts simultaneously:
+
+1. `L_core/w` saturates near the #758 optimizer;
+2. the excess `L-L_core` carries the linear tail cost;
+3. `U` approaches Uniform(0,1) away from zero-excess cases;
+4. the scalar span spectrum develops a double-pole/Erlang prefactor.
+
+Failure of (4.1), especially strong endpoint condensation, would favor an asymmetric single-long-branch mechanism and would require a different explanation of the opposite residues.
+
+## 5. Split poles as a finite-width deformation
+
+The empirical poles are close but not exactly equal. A natural finite-width deformation is
+
+```text
+gamma_- = gamma_bar - Delta/2,
+gamma_+ = gamma_bar + Delta/2.                            (5.1)
+```
+
+Then the branch-sum coefficient is the hypoexponential form (1.2). The near-equal masses may come from weak endpoint/core asymmetry or from interaction between the two branch excursions through the finite core.
+
+This interpretation is distinct from, but not mutually exclusive with, a metastable two-sector transfer block. The decisive check is the actual slow eigenvector geometry:
+
+- if the two modes localize on upper/lower branch or two serial excursion stages, the convolution mechanism is supported;
+- if they instead distinguish a global topology bit unrelated to branch orientation, the metastable-sector mechanism is more plausible.
+
+## 6. Relation to the loop--branch rate
+
+Above the saturated core size `r_*(p)`, the #758 candidate is
+
+```text
+I_p(A)=kappa A + c_*,
+```
+
+so all partitions of the excess span between the two ends have the same leading exponential cost. The continuum of branch splits is exactly the kind of zero mode that produces a polynomial prefactor while leaving the large-deviation rate unchanged.
+
+A sharpened conjecture is therefore:
+
+```text
+P_Palm(L≈A w)
+ = w^beta C_p(A) exp[-w I_p(A)] [1+o(1)],                (6.1)
+```
+
+with an additional `beta=1` contribution from the two-ended split degeneracy in the strictly saturated linear branch, relative to a convention where the core location and one branch split are otherwise fixed. The total exponent beta also contains rooting/unrooting and transverse fluctuation factors, so `beta=1` is **not** claimed as the final complete-component prefactor.
+
+The robust part is the predicted linear-in-excess split measure and Uniform(0,1) conditional fraction, not a final absolute power of w.
+
+## 7. What to test next
+
+### Existing operator, no new width production
+
+Extract the two slow visible eigenvectors and inspect whether their state mass distinguishes upper/lower branch stages or a global topology bit.
+
+### #762 morphology
+
+At `L>=A w` for `A=1,2`, record the already planned
+
+```text
+L_core,
+A_-, A_+,
+U=A_-/(A_-+A_+).
+```
+
+The strongest mechanism test is not the mean of U but its conditional law at fixed excess-span bins.
+
+### Spectrum
+
+Fit the late sequence directly to both models:
+
+```text
+M1: c1 rho1^h + c2 rho2^h,
+M2: (a+b h) rho^h,
+```
+
+on nested windows. If `M2` becomes competitive as w grows while the fitted pole split shrinks, that is the expected coalescing-convolution signature.
+
+## 8. Claim boundary
+
+- Exact algebra: convolution of two simple poles gives opposite residues and a double-pole/Erlang limit; equal-rate exponentials give the Uniform branch fraction.
+- Data-supported hypothesis: the SITE span doublet may be realizing this two-ended branch convolution.
+- Not claimed: asymptotic independence of `A_-` and `A_+`, identification of the slow eigenvectors, or a final complete-component w-prefactor.
diff --git a/docs/manuscripts/geometric-balance/span-visible-pole-splitting-20260914.md b/docs/manuscripts/geometric-balance/span-visible-pole-splitting-20260914.md
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+# Coalescing visible poles in the complete-component span spectrum
+
+Date: 2026-09-14
+
+Status: bounded diagnostic from the already committed `span-spectrum-20260913.json`, plus a correction to one proposed proof route for #758/#760. No new transfer build or width production is used.
+
+## 1. Why this check matters
+
+For fixed circumference `w`, the tagged complete-component span sequence has a finite-state representation
+
+```text
+d_h = alpha R^(h-1) b.
+```
+
+Hence its ultimate `h->infinity` decay is controlled by the slowest **visible** pole of the source/readout pair. A tempting bridge between #758 and #760 is to assume that the relaxation to that pole occurs on a scale `o(w)`, so that the simultaneous macroscopic regime `h=A w` already sees the fixed-w Perron tail.
+
+The existing span data give a strong negative diagnostic for that assumption.
+
+## 2. Frozen two-pole extraction
+
+Input:
+
+```text
+results/geometric-consistency/span-spectrum-20260913.json
+```
+
+No solver is rerun.
+
+For each stored `d_h` sequence, fit on the late window beginning at `h=13` the second-order recurrence
+
+```text
+d_(h+2) = S d_(h+1) - P d_h.                              (2.1)
+```
+
+Each row is divided by `d_(h+1)` before the two-parameter least-squares solve, avoiding the many-decade dynamic range. The two recurrence roots are reported as effective visible poles
+
+```text
+rho_1 > rho_2 > 0.
+```
+
+A separate two-exponential fit
+
+```text
+d_h ~= c_1 rho_1^(h-1) + c_2 rho_2^(h-1)                 (2.2)
+```
+
+checks the residues. This is a projected/Prony diagnostic, not a certified full-matrix eigensolve.
+
+For NN `p=1/4`, widths 2--7 give stable real roots and late-window relative residuals between about `1e-12` and `1e-7` (the largest widths inherit the stored float residuals).
+
+## 3. NN p=1/4 result
+
+Representative values are:
+
+| w | rho1 | rho2 | Delta gamma=log(rho1/rho2) | xi_rel=1/Delta gamma | xi_rel/w | c2/c1 |
+|---:|---:|---:|---:|---:|---:|---:|
+| 2 | .290359457 | .187500000 | .437340815 | 2.29 | 1.14 | -.9984 |
+| 3 | .329288918 | .303004745 | .083187071 | 12.02 | 4.01 | -1.0049 |
+| 4 | .344536629 | .334361548 | .029977523 | 33.36 | 8.34 | -1.0204 |
+| 5 | .350913754 | .346766728 | .011888178 | 84.12 | 16.82 | -1.0153 |
+| 6 | .353568560 | .351836957 | .004909537 | 203.69 | 33.95 | -1.0091 |
+| 7 | .354597156 | .354034203 | .001588845 | 629.39 | 89.91 | -1.0038 |
+
+Two features are much more important than the individual digits:
+
+1. the two slow visible poles rapidly coalesce;
+2. their fitted residues are nearly equal and opposite.
+
+Thus the scalar span response is not approaching its one-pole regime on an `O(w)` height scale in these controls.
+
+The same phenomenon becomes even stronger deeper in the subcritical phase / on the matching graph. Examples from the same stored file:
+
+```text
+NN p=1/8, w=6:        xi_rel ~= 5288,  c2/c1 ~= -1.00026;
+matching p=1/16, w=5: xi_rel ~= 321,   c2/c1 ~= -1.00365;
+matching p=1/16, w=6: xi_rel ~= 1209,  c2/c1 ~= -1.00138.
+```
+
+So this is not a one-point anomaly of NN `p=1/4`.
+
+## 4. Direct macroscopic-hazard check
+
+Let
+
+```text
+T_w(h)=sum_{j>=h} d_j
+```
+
+with the stored tail bin appended. Define the finite-height hazard
+
+```text
+gamma_eff,w(h) = -log[T_w(h+1)/T_w(h)].                   (4.1)
+```
+
+For NN `p=1/4`:
+
+```text
+w=4: gamma_eff(w)=0.7790, gamma_eff(2w)=0.9436,
+     ultimate two-pole gamma1=1.0656;
+w=5: 0.7867, 0.9388, gamma1=1.0472;
+w=6: 0.8042, 0.9440, gamma1=1.0397;
+w=7: 0.8242, 0.9519, gamma1=1.0368.
+```
+
+The `h=w,2w` tails are visibly not in the fixed-w ultimate Perron regime.
+
+This does **not** falsify the #758 macroscopic LDP. It falsifies the shortcut that would derive that LDP by first taking `h->infinity` at fixed w and then assuming a relaxation length `o(w)`.
+
+## 5. Noncommuting limits and a replacement asymptotic picture
+
+Write the two slow pole masses as
+
+```text
+gamma_1 = gamma_bar - Delta/2,
+gamma_2 = gamma_bar + Delta/2,
+```
+
+and suppose the corresponding residues are approximately `A` and `-A`, plus a smaller common component `B/2`. Then
+
+```text
+d_h
+ ~= exp(-gamma_bar h)
+    [ B cosh(Delta h/2) + 2 A sinh(Delta h/2) ].           (5.1)
+```
+
+If `Delta w -> 0`, then for `h=A0 w`
+
+```text
+d_h
+ ~= exp(-gamma_bar h)
+    [ B + A Delta h + o(Delta h) ].                       (5.2)
+```
+
+Thus a coalescing pair can have an exponentially long fixed-w relaxation time while the simultaneous `h=O(w)` exponential rate is still governed by the common centre mass `gamma_bar`. The near-cancellation changes subexponential/polynomial factors much more strongly than the large-deviation exponent.
+
+This is a better interface for #758:
+
+> prove that all macroscopically relevant visible slow modes have masses converging to the same planar cost `kappa`, while all truly fast modes remain separated; do **not** require the slow doublet to have mixed to one Perron vector by height `A w`.
+
+## 6. Relation to #760
+
+The periodic-locality argument on #760 concerns the location of the ultimate cylinder mass `gamma_w` relative to the plane mass `kappa`. The coalescing-pole diagnostic does not contradict an exponentially small `gamma_w-kappa` correction.
+
+What it changes is measurement/proof strategy:
+
+- finite-height hazards at `h=O(w)` should not be used as direct estimators of the ultimate fixed-w `gamma_w`;
+- conversely, the fixed-w leading Perron pole should not be substituted into an `h=A w` morphology argument without controlling the neighboring visible pole and its residue.
+
+The two limits
+
+```text
+h->infinity at fixed w,
+and
+w->infinity with h=A w
+```
+
+need not commute at the level of the visible decomposition even if both exponential masses converge to `kappa`.
+
+## 7. A new metastable-doublet conjecture
+
+For fixed subcritical p, the data suggest a two-dimensional slow visible subspace whose splitting is exponentially small in w. A deliberately weak conjecture is
+
+```text
+Delta gamma_w = exp[-Theta(w)],                            (7.1)
+```
+
+with opposite-sign source/readout residues and a common centre mass tending to the plane connection mass.
+
+For NN `p=1/4`, the observed splitting is numerically comparable to `w nu_w`:
+
+```text
+Delta gamma_w/(w nu_w)
+ = 1.83, 1.69, 1.77, 1.94, 1.67     for w=3,...,7.
+```
+
+At other p/graphs the proportionality constant is very different, so no universal constant is claimed. A more plausible structural version is
+
+```text
+Delta gamma_w = Theta_p(essential-component coverage/renewal intensity),
+```
+
+with `w nu_w` only a first proxy.
+
+A possible mechanism is a metastable two-sector transfer block: rare complete-topology events weakly couple two otherwise degenerate longitudinal sectors, producing a symmetric/antisymmetric splitting and the observed opposite residues. The physical sectors should be identified from the actual tagged states before this interpretation is promoted.
+
+## 8. What should be checked next
+
+1. Extract the two leading **visible** poles directly from the tagged resolvent/operator, with left/right source overlaps, rather than only from `d_h`.
+2. Identify which tagged frontier states carry the two slow eigenvectors.
+3. Compare `Delta gamma_w` to `nu_w E_Palm K`, `nu_w E_Palm L`, and other renewal/coverage intensities once #774 supplies the relevant moments.
+4. Repeat the projected recurrence on the stored matching `p=1/8` sequence and any future width-8 span spectrum without rebuilding the transfer.
+5. For #758, formulate the upper-bound/LDP proof directly in the simultaneous `h=A w` limit; the coalescing pair explains why a fixed-w single-pole shortcut is unsafe.
+
+## 9. Claim boundary
+
+- Data statement: the two-pole recurrence and residue diagnostics on the committed span spectra.
+- Strong negative diagnostic: `xi_rel=o(w)` is incompatible with the observed width trend and should not be used as a proof assumption without a new mechanism that reverses it.
+- Conjecture: exponential slow-mode coalescence / metastable two-sector interpretation and its relation to renewal intensity.
+- Not claimed: a certified full transfer eigendecomposition, an asymptotic exponent for the splitting, or a modification of the #758 rate function.
diff --git a/results/geometric-consistency/span-visible-pole-splitting-20260914.json b/results/geometric-consistency/span-visible-pole-splitting-20260914.json
new file mode 100644
index 000000000..88051d5a4
--- /dev/null
+++ b/results/geometric-consistency/span-visible-pole-splitting-20260914.json
@@ -0,0 +1,27 @@
+{
+  "schema": "matching-one/span-visible-pole-splitting/v1",
+  "date": "2026-09-14",
+  "source_path": "results/geometric-consistency/span-spectrum-20260913.json",
+  "source_sha": "58e070f814f290d1d6a7d10c00ab4273d671c5bb",
+  "method": "normalized second-order late recurrence starting at h=13 plus separate two-exponential residue fit",
+  "claim_boundary": "Projected/Prony diagnostic on an existing scalar sequence; not a certified full transfer eigendecomposition.",
+  "nn_p_1_4": [
+    {"width":2,"rho1":0.29035945694,"rho2":0.18750000038,"delta_gamma":0.437340815,"relaxation_length":2.2865,"relaxation_length_over_width":1.143,"c2_over_c1":-0.9984},
+    {"width":3,"rho1":0.32928891833,"rho2":0.30300474518,"delta_gamma":0.083187071,"relaxation_length":12.021,"relaxation_length_over_width":4.007,"c2_over_c1":-1.00485,"delta_gamma_over_w_nu":1.8256},
+    {"width":4,"rho1":0.34453662879,"rho2":0.33436154796,"delta_gamma":0.029977523,"relaxation_length":33.358,"relaxation_length_over_width":8.340,"c2_over_c1":-1.02036,"delta_gamma_over_w_nu":1.6919,"gamma_eff_at_w":0.7790,"gamma_eff_at_2w":0.94357,"gamma1":1.06555},
+    {"width":5,"rho1":0.35091375383,"rho2":0.34676672776,"delta_gamma":0.011888178,"relaxation_length":84.117,"relaxation_length_over_width":16.823,"c2_over_c1":-1.01527,"delta_gamma_over_w_nu":1.7687,"gamma_eff_at_w":0.78670,"gamma_eff_at_2w":0.93879,"gamma1":1.04721},
+    {"width":6,"rho1":0.35356856032,"rho2":0.35183695653,"delta_gamma":0.004909537,"relaxation_length":203.685,"relaxation_length_over_width":33.948,"c2_over_c1":-1.00910,"delta_gamma_over_w_nu":1.9391,"gamma_eff_at_w":0.80418,"gamma_eff_at_2w":0.94399,"gamma1":1.03968},
+    {"width":7,"rho1":0.35459715576,"rho2":0.35403420318,"delta_gamma":0.001588845,"relaxation_length":629.388,"relaxation_length_over_width":89.913,"c2_over_c1":-1.00377,"delta_gamma_over_w_nu":1.6737,"gamma_eff_at_w":0.82422,"gamma_eff_at_2w":0.95189,"gamma1":1.03677}
+  ],
+  "secondary_controls": [
+    {"label":"NN p=1/8","width":6,"rho1":0.15331177010,"rho2":0.15328278242,"delta_gamma":0.00018909458,"relaxation_length":5288.36,"relaxation_length_over_width":881.39,"c2_over_c1":-1.000264,"max_recurrence_relative_residual":6.2e-11},
+    {"label":"matching p=1/16","width":5,"rho1":0.18310566243,"rho2":0.18253656178,"delta_gamma":0.003112885,"relaxation_length":321.25,"relaxation_length_over_width":64.25,"c2_over_c1":-1.00365},
+    {"label":"matching p=1/16","width":6,"rho1":0.18355402843,"rho2":0.18340223624,"delta_gamma":0.000827304,"relaxation_length":1208.75,"relaxation_length_over_width":201.46,"c2_over_c1":-1.00138}
+  ],
+  "verdicts": {
+    "single_pole_relaxation_o_w": "incompatible with current width trend",
+    "macroscopic_LDP_candidate": "not falsified",
+    "periodic_mass_locality_gamma_minus_kappa": "not falsified",
+    "recommended_interface": "coalescing slow visible subspace; analyze h=A w directly rather than assuming fixed-w Perron capture"
+  }
+}
diff --git a/scripts/span_visible_pole_splitting.py b/scripts/span_visible_pole_splitting.py
new file mode 100644
index 000000000..18952dd08
--- /dev/null
+++ b/scripts/span_visible_pole_splitting.py
@@ -0,0 +1,80 @@
+#!/usr/bin/env python3
+"""Projected two-pole diagnostic for complete-component span spectra.
+
+Reads the already generated span-spectrum JSON and does not rebuild a transfer
+matrix. Fits d_{h+2}=S d_{h+1}-P d_h on a declared late window after dividing
+each row by d_{h+1}, then fits the two exponential residues.
+"""
+from __future__ import annotations
+import argparse, json, math
+from pathlib import Path
+
+
+def solve_2x2(a11,a12,a22,b1,b2):
+    det=a11*a22-a12*a12
+    if abs(det)<1e-30: raise ArithmeticError("singular normal equations")
+    return ((b1*a22-b2*a12)/det,(a11*b2-a12*b1)/det)
+
+
+def fit_recurrence(d,start_h=13):
+    rows=[]
+    for h in range(start_h,len(d)-1):
+        a,b,c=d[h-1],d[h],d[h+1]
+        if min(a,b,c)<=0: continue
+        rows.append((1.0,-a/b,c/b,h))
+    if len(rows)<4: raise ValueError("too few late-window triples")
+    a11=sum(x*x for x,z,y,h in rows); a12=sum(x*z for x,z,y,h in rows); a22=sum(z*z for x,z,y,h in rows)
+    b1=sum(x*y for x,z,y,h in rows); b2=sum(z*y for x,z,y,h in rows)
+    S,P=solve_2x2(a11,a12,a22,b1,b2)
+    disc=S*S-4*P
+    if disc<=0: raise ArithmeticError("nonpositive recurrence discriminant")
+    rr=math.sqrt(disc); rho1=(S+rr)/2; rho2=(S-rr)/2
+    if not (rho1>rho2>0): raise ArithmeticError("unexpected recurrence roots")
+    rec=[]
+    for x,z,y,h in rows:
+        pred=S+z*P; rec.append(abs(pred-y)/abs(y))
+
+    q=rho2/rho1; xs=[]; ys=[]; hs=[]
+    for h in range(start_h,len(d)+1):
+        val=d[h-1]
+        if val<=0: continue
+        xs.append(q**(h-1)); ys.append(val/(rho1**(h-1))); hs.append(h)
+    n=len(xs); sx=sum(xs); sxx=sum(x*x for x in xs); sy=sum(ys); sxy=sum(x*y for x,y in zip(xs,ys))
+    c1,c2=solve_2x2(float(n),sx,sxx,sy,sxy)
+    fit=[]
+    for h in hs:
+        val=d[h-1]; pred=c1*rho1**(h-1)+c2*rho2**(h-1); fit.append(abs(pred-val)/val)
+    dg=math.log(rho1/rho2)
+    return {"start_h":start_h,"rho1":rho1,"rho2":rho2,"gamma1":-math.log(rho1),"gamma2":-math.log(rho2),
+            "delta_gamma":dg,"relaxation_length":1/dg,"c1":c1,"c2":c2,"c2_over_c1":c2/c1,
+            "max_recurrence_relative_residual":max(rec),"max_two_exp_relative_residual":max(fit)}
+
+
+def tail_hazard(d,tail,h):
+    if h<1 or h>=len(d)+1: return None
+    t0=sum(d[h-1:])+tail; t1=sum(d[h:])+tail
+    return -math.log(t1/t0) if t0>0 and t1>0 else None
+
+
+def run_record(rec,start_h):
+    d=[float(x) for x in rec["d_h_float"]]; fit=fit_recurrence(d,start_h); w=int(rec["width"]); nu=float(rec.get("nu_total_float",sum(d)))
+    return {"label":rec.get("label"),"width":w,"matching":bool(rec.get("matching")),"p":rec.get("p"),"d_max":len(d),"nu":nu,**fit,
+            "relaxation_length_over_width":fit["relaxation_length"]/w,
+            "delta_gamma_over_w_nu":fit["delta_gamma"]/(w*nu) if nu>0 else None,
+            "gamma_eff_at_w":tail_hazard(d,float(rec.get("tail_bin_float",0.0)),w),
+            "gamma_eff_at_2w":tail_hazard(d,float(rec.get("tail_bin_float",0.0)),2*w)}
+
+
+def main():
+    ap=argparse.ArgumentParser(); ap.add_argument("--input",required=True); ap.add_argument("--start-h",type=int,default=13); ap.add_argument("--output")
+    a=ap.parse_args(); payload=json.loads(Path(a.input).read_text()); rows=[]; errors=[]
+    for rec in payload["runs"]:
+        try: rows.append(run_record(rec,a.start_h))
+        except Exception as exc: errors.append({"label":rec.get("label"),"width":rec.get("width"),"matching":rec.get("matching"),"p":rec.get("p"),"error":str(exc)})
+    out={"schema":"matching-one/span-visible-pole-splitting/v1","source":a.input,"start_h":a.start_h,
+         "method":"normalized second-order recurrence + two-exponential residue fit","rows":rows,"errors":errors,
+         "claim_boundary":"Projected/Prony diagnostic only; not a certified full transfer eigendecomposition."}
+    txt=json.dumps(out,indent=2)
+    if a.output: Path(a.output).write_text(txt+"\n")
+    print(txt)
+if __name__=="__main__": main()