analysis: span spectrum shows a coalescing effective slow doublet

docs/manuscripts/geometric-balance/branch-endpoint-dressing-beta-law-20260914.md

@@ -0,0 +1,157 @@
+# 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.

docs/manuscripts/geometric-balance/span-double-pole-branch-convolution-20260914.md

@@ -0,0 +1,212 @@
+# 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.