Thin-torus two-birth limits and source modes (stacked on #710; do not merge)

notes/thin-torus-central-layer-source-20260912.md

@@ -0,0 +1,209 @@
+# What the cylinder balance root resolves: an exponentially rare conditional law
+
+2026-09-12. Completed conditional spectral analysis, instantiated by the proved
+width-2/3/4 square-site formulas. Connects #636, #276, #337 and #582. It is
+separate from the all-fixed-width geometric endpoint theorem in
+`thin-torus-two-birth-limits-20260912.md`.
+
+## 1. Local spectral hypotheses, not an all-width assumption
+
+Suppose on a neighborhood of an interior q, at a FIXED width,
+
+    P_0(p;m)=lambda_0(p)^m[1+O(eta^m)],
+    P_2(p;m)=lambda_2(p)^m[1+O(eta^m)],                  (1)
+
+uniformly there, with the required derivative bounds, 0<eta<1, analytic positive
+simple leading branches, and
+
+    lambda_0(q)=lambda_2(q)=lambda_* in (0,1),
+    a_0=(log lambda_0)'(q)<0<a_2=(log lambda_2)'(q).
+
+Leading coefficients are one in these formulas. This is justified locally by
+the already proved width-2/3/4 event/spectral representations. It is NOT assumed
+at every width. For general nonunit leading coefficients the formulas below
+acquire the corresponding amplitude ratio; equality of exponents alone does
+not erase it.
+
+At width four the preceding rational spectral certificate bounds the common
+subleading block below the two leading eigenvalues. Its relative Perron value
+at q is about .773935. This block cancels in M but not in each probability.
+Thus it may be used for the hypotheses (1), while the much faster M correction
+uses a different observable rate. This distinction survives source tilting.
+
+## 2. A sharp conditional transition inside a flat unconditional CDF
+
+Set p=q+x/m for x in a fixed bounded set. Expanding the logarithms gives
+
+    P_j(q+x/m;m)/lambda_*^m -> exp(a_j*x),
+    M(q+x/m;m)/lambda_*^m -> g(x)
+                             := exp(a_2*x)-exp(a_0*x).  (2)
+
+The unconditional threshold CDF remains F=(1+M)/2 ->1/2. In contrast,
+
+    P(r=2 | r!=1, p=q+x/m)
+      = P_2/(P_0+P_2)
+      -> 1/[1+exp(-(a_2-a_0)*x)].                      (3)
+
+Equivalently, the conditional signed contrast M/E tends to
+`tanh((a_2-a_0)*x/2)`, where E=P_0+P_2. Formula (3) is a rare-sector conditional
+law. It is not the distributional threshold law F and not a claim that most
+configurations undergo a critical transition at q.
+
+This gives the precise object that a fixed-width spectral crossing organizes:
+the relative costs of the two exponentially rare alternatives to rank one.
+The sigmoid and the flat CDF coexist because they refer to different measures.
+
+## 3. Quantile probability window and inverse conditioning
+
+Since a_0<0<a_2, g is strictly increasing from -infinity to +infinity. For fixed
+real z define
+
+    u_m = 1/2 + (z/2)*lambda_*^m.
+
+For all sufficiently large m this is an interior probability level. From (2),
+monotonicity and locally uniform convergence,
+
+    m [Q_m(u_m)-q] -> g^-1(z).                         (4)
+
+The p window is of order 1/m, but its probability mass is exponentially small,
+of order lambda_*^m. Every FIXED nonmedian quantile lies outside it and instead
+obeys the endpoint laws proved in the companion note.
+
+Let h'=a_2-a_0. At the finite matching median p_m=q+o(1/m),
+
+    M'(p_m) ~ m*h'*lambda_*^m,
+    Q_m'(1/2) ~ 2/[m*h'*lambda_*^m].                   (5)
+
+Thus the inverse CDF becomes exponentially ill-conditioned at its median even
+though that median's deterministic spectral bias can be exponentially small.
+Calling a shrinking finite-root bias 'increased statistical resolution' would
+be wrong here.
+
+At width four, the preceding finite root theorem gives
+
+    q = .591417170853138481798834101735923177964270443...,
+    lambda_* = .844850251961768087272984426486720187657...,
+    a_0 = -1.215224209362996573970582674...,
+    a_2 =  1.249128400110595642565560891...,
+    h'  =  2.464352609473592216536143565....
+
+These are deterministic high-precision diagnostics of the supplied finite
+polynomials. The q value is already in Jacobsen 2015 Table 2. None is a new
+infinite-square critical probability.
+
+## 4. Source balance: exact finite formula and its limit
+
+Use the intrinsic rank source from #337, X=r-1, not a Potts-Q continuation:
+
+    Z(p,s)=P_0 e^-s + P_1 + P_2 e^s,
+    M_s(p)=partial_s log Z
+           = [P_2 e^s-P_0 e^-s]/Z.                    (6)
+
+For every finite torus and real s, the unique source-balanced probability p_m(s)
+is determined by
+
+    log[P_2(p_m(s))/P_0(p_m(s))]+2s=0.                (7)
+
+The normalizer is positive and cancels from the zero condition; it does NOT
+cancel from arbitrary responses. P_2 is strictly increasing and P_0 strictly
+decreasing on the interior, so the ratio runs continuously from zero to infinity.
+
+Equations (1) and (7) imply for each bounded fixed s,
+
+    p_m(s)=q-2s/[m*h']+O(m^-2+eta^m/m).               (8)
+
+The exact finite differential identity at s=0 is
+
+    d p_m(s)/ds |_(0) = -E(p_m)/M'(p_m)
+                        ~ -2/(m*h').                 (9)
+
+For width four, `m*dp_m/ds -> -.8115721720631601937...`.
+The numerical reader solves (7) using the full finite probabilities at s=+/-1/2
+and m=16,64,256, without substituting the limiting formula into the finite answer.
+For example m=256 gives m(p_m(.5)-q)=-.405935683764... versus the limit
+-.405786086031..., and the opposite source gives +.405630373026....
+
+An important scope point: at s=0 the common B16 trace cancels from M, but at
+nonzero s the unnormalized numerator is
+
+    e^s tr(B15^m)-e^-s tr(B5^m)
+      -2 sinh(s) tr(B16^m)+4 sinh(s) t^(2m),           (10)
+
+before division by (1+t)^(4m). The slower shared mode is restored. The original
+s=0 remainder rate therefore cannot silently be used for all source responses.
+No locality or continuum-field interpretation of s is claimed.
+
+## 5. A large-deviation source limit, without calling it criticality
+
+For fixed p in the neighborhood where (1) holds, set s=m*sigma. Since P_1->1,
+log-sum-exp gives
+
+    lim_m (1/m) log Z(p,m*sigma)
+       = max{0, log lambda_0(p)-sigma,
+                log lambda_2(p)+sigma}.              (11)
+
+At q, the two activation points are +/-[-log lambda_*]. For bounded unscaled s,
+Z tends simply to 1, as the geometric theorem already proves. The piecewise
+linear limit (11) describes reweighting exponentially rare rank sectors, not
+the original iid percolation measure's two-dimensional phase transition.
+
+In particular, the equality of the two rare-event rates at q has real content,
+while the unscaled source symmetry Z->1 holds throughout the thin-cylinder
+interior and has no threshold-identifying content on its own.
+
+## 6. Statistical consequence — deliberately restricted to one estimator
+
+Suppose n independent Bernoulli configurations are sampled at ONE p, and only
+X=r-1 is measured per configuration. Then exactly
+
+    Var(X)=E-M^2,
+    P(all n observations equal zero)=(1-E)^n.         (12)
+
+To see at least one nonzero X with probability at least .95 requires
+
+    n >= ceil[log(.05)/log(1-E)].                      (13)
+
+This is only a one-event criterion, NOT enough to estimate a root accurately.
+In the local Gaussian inverse-response regime, with nE large,
+
+    se(p_est) approximately sqrt(E-M^2)/(sqrt(n)*M'),
+    se(p_est) at balance ~ sqrt(2)/[sqrt(n)*m*h'*lambda_*^(m/2)]. (14)
+
+For nE not large, a normal root-error reference is not justified. At q4 the
+full finite formulas give the following numerical diagnostics:
+
+| m | E(q4) | approximate n for 95% chance of one nonzero X |
+|---:|---:|---:|
+| 8 | .45227368437 | 5 |
+| 32 | .009075329013 | 329 |
+| 128 | 8.488607461e-10 | 3.529120985e9 |
+| 512 | 6.490166177e-38 | 4.615802110e37 |
+
+These rows use q4, NOT the exact finite p_m; the difference is immaterial to
+the asymptotic but is recorded rather than hidden. Numerical sample values are
+not interval-certified integer ceilings. The formula (13) is exact, while the
+listed inputs and rounded evaluations are high-precision diagnostics.
+
+This is not an algorithm-independent lower bound. It does NOT apply unchanged
+to Newman-Ziff permutation histograms, conditional/binomial reconstruction,
+Rao-Blackwell estimators, rare-event importance sampling, or exact transfer
+algebra. In particular the existing N145/N290/N725 productions are different
+geometries and different estimators. No new acquisition is priced or requested.
+
+## 7. Numerical precision and executed result
+
+`thin_torus_spectral_controls.py` evaluates only small companion-matrix powers,
+so m=1,000,000 is NOT a million-row simulation. All outputs are numerical
+controls, not replacements for the proofs. At 80 decimal digits, three probes
+per width are re-evaluated at 110 digits; the maximum absolute difference in M
+is below 1e-70. This checks numerical precision, not independent physics.
+
+At q and large m the true residual is below the working precision relative to
+the two leading terms. Such M entries are explicitly null/unresolved, rather
+than interpreting subtraction noise as a physical finite-length correction.
+Positive P_0/P_2 and their activity remain separately evaluated in this regime.
+
+The output retains endpoint expansions, central profiles, rare-sector logistic
+values, full source-balance roots, source large-deviation checks, quartiles,
+and the narrowly defined snapshot-cost diagnostics. No fitted exponent or
+sampling budget enters any of these calculations.

notes/thin-torus-source-handoff-20260912.md

@@ -0,0 +1,71 @@
+# Completed analysis for existing #613/#276/#582 and #337/#636
+
+2026-09-12. This is a handoff for existing issues, not a request for new production.
+No remote issue lifecycle or PR state was changed by this local delivery.
+
+## For #613 / #276 / #582: completed thin-geometry boundary
+
+The exact finite row inequalities P0 <= (1-p^w)^m and
+P2 <= [1-(1-p)^w]^m give rank-one concentration on fixed-width long tori and
+on w=o(log m). The threshold CDF F=(1+M)/2 tends to 1/2 throughout (0,1),
+while its probability law tends to (delta_0+delta_1)/2. In particular,
+'w and m both grow' is not a sufficient replacement for #613's geometric
+hypothesis: w_j=j, m_j=ceil(exp(j^2)) is a counterexample. The stated
+ell/log N -> infinity theorem is NOT contradicted.
+
+For every fixed w>=2 the two births have an independent limiting endpoint
+law, with survival exp(-x^w) and exp(-c_w y^w), where
+c_w=[z^0](z^-1+1+z)^w. The proof classifies minimal motifs and controls
+nonminimal circuits and dependent rare-pattern counts. Joint moments and
+permutation birth-rank clocks are covered. No growing-width Weibull uniformity
+is asserted. Do not commission another small census or a thin-torus production
+merely to rediscover this limit.
+
+For widths 2/3/4 the spectral crossing is an internal balance of two rare
+sectors, not concentration of the entire threshold law. The conditional
+rank-2 probability is logistic on p=q+x/m, while F remains flat. The probability
+window around the median is exponentially small. Its inverse sensitivity is
+not improved by the exponentially small deterministic root bias.
+
+Files: notes/thin-torus-two-birth-limits-20260912.md;
+notes/thin-torus-central-layer-source-20260912.md; the two thin_torus scripts
+and their generated results.
+
+## For #337 / #636: completed source-visible spectrum
+
+Using the preceding all-parameter P0/P2 trace identities and #708's original
+509-state certificate, the generic scalar orders are M:16, E=P0+P2:28,
+twice the tilted-mean numerator at exp(s)=2:28, twice Z at exp(s)=2:29.
+Full integer Hankel minors and independent modular residues prove the lower
+bounds; existing exact trace identities prove the upper bounds for all lengths.
+The normalized tilted mean is a ratio, NOT claimed to satisfy an order-28
+constant-coefficient linear recurrence.
+
+A bounded intrinsic source restores common modes canceled at s=0. Its balance
+root has exact derivative -E/M', and, under the proved local spectral contract,
+shift -2s/(m*h')+O(m^-2+eta^m/m). For the thin sequences, Z(p,s)->1 everywhere
+inside (0,1) at bounded s, so restored source symmetry alone is not a critical
+point identifier. The source algebra remains correct; this is a scope correction
+for interpreting it.
+
+Files: notes/width4-topological-source-visible-modes-20260912.md;
+scripts/width4_topological_source_spectrum.py; full source-spectrum JSON.
+
+## Allocation and limits
+
+No new task is needed for these completed analyses. No GPU, wider state table,
+new Monte Carlo, or large retrieval is justified by this delivery alone.
+
+The remaining cross-width question needs estimates uniform in width on the
+VISIBLE mode weights and gaps, plus geometry and readout conventions. Neither
+a fixed-width theorem nor a generic source-order table provides that uniformity.
+A future computation should target one stated separating inequality, not a
+width ladder for its own sake. #275's same-source original-U prediction columns
+remain a distinct missing physical input; no candidates are promoted here.
+
+The snapshot one-nonzero-event cost formula is restricted to iid rank
+observations at one p. It is not a budget for Newman-Ziff histogram reconstruction,
+Rao-Blackwellization, importance sampling, or exact transfer algebra.
+
+The preceding parametric definition is an explicit dependency and is not
+asserted merged. Existing source and production artifacts are untouched.