Audit scale-dependent seed averaging in the full recurrence

research/scale-dependent-seed-moment-audit/PROOF.md

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+# Scale-dependent seeds require the complete recurrence
+
+This is a written proof about a positive recursive cost model. It is not a
+physical supplier construction, a multiplication bound or a formal-kernel proof.
+All displayed child multiplicities are charged. Glue, adapters and other work
+are set to zero, a favorable relaxation of a complete implementation.
+
+## 1. The precise source implication being tested
+
+Philippe Van Velzen's `limits-1/single-letter` lane at commit
+`ed21d6245f8ce1a62e21205be7bedc24271dd772` considers
+
+    T(x) = sum_r (n_r/W) T(rx/m) + h(x),   T(x)=1 for 0<x<=1.
+
+A seed has scalar root p satisfying `sum_r (n_r/W)(r/m)^p=1`, and supplier
+saving a=1-p. Its Theorem 1 keeps an explicit cumulative local-oscillation
+error when comparing T with `exp(integral p d log x)`. The following
+unqualified Cesaro corollary/summary does not follow unless that error and the
+normalized glue contribution are sublinear in log x. This audit tests that
+implication, not the full error-qualified inequality or the other source lanes.
+
+## 2. An irreducible, two-rank counterexample
+
+Take common width m=256 and stock W=1. There are two complete profiles:
+
+| seed | n_16 | n_1 | exact scalar root p | a=1-p |
+|---|---:|---:|---:|---:|
+| A | 1 | 2 | 1/4 | 3/4 |
+| B | 4 | 32 | 3/4 | 1/4 |
+
+Indeed the two terms in each seed's moment at its root are exactly 1/2.
+These are positive integer multiplicities, with child ratios 1/16 and 1/256.
+No claim is made that these profiles have native supplier realizations.
+A further restriction on realizable profiles or admissible seed variation
+could exclude this witness; it would need to be an explicit hypothesis.
+
+For x>1 put i=ceil(log_16 x). Use A for odd i and B for even i, with h=0.
+Multiplication by either child ratio subtracts one or two from i, including
+at exact interval endpoints. Thus T is constant on each scale interval and
+equals t_i, where t_i=1 for i<=0 and
+
+    t_i = t_(i-1) + 2t_(i-2)       (i odd),
+    t_i = 4t_(i-1) + 32t_(i-2)     (i even).
+
+The first values are t_0=1,t_1=3,t_2=44,t_3=50,t_4=1608. Every child call
+is included. The local root function alternates between 1/4 and 3/4 on
+equal log-length intervals. Its log-Cesaro mean therefore tends to 1/2.
+
+Let v_k=(t_(2k),t_(2k-1))^T, interpreting t_(-1)=1. Then
+
+    v_k = P^k (1,1)^T,     P = [[36,8],[1,2]].
+
+The characteristic polynomial is X^2-38X+64. Its dominant eigenvalue is
+lambda=19+3 sqrt(33), with positive eigenvector w=(lambda-2,1)^T.
+Since lambda-2>1,
+
+    w/(lambda-2) <= (1,1)^T <= w.
+
+P has nonnegative entries, so multiplication by P^k preserves the inequalities.
+The first coordinate gives
+
+    lambda^k <= t_(2k) <= (lambda-2)lambda^k.
+
+The odd subsequence is a positive fixed linear combination of adjacent even
+state coordinates and has the same exponential rate. Consequently, for all
+x>=1,
+
+    T(x) = Theta(x^p_eff),    p_eff = log(lambda)/log(256).
+
+This is an all-scale proof, with constants independent of x. In particular,
+32<lambda<37: the polynomial at 32 is -128, its larger root lies to the
+right of 19, and sqrt(33)<6. Since 37^3<256^2, therefore
+
+    5/8 < p_eff < 2/3,      1/3 < a_eff < 3/8.
+
+The claimed log-Cesaro substitution would instead give p_eff=1/2 and a=1/2.
+The discrepancy persists in the exponent. It is not a numerical rounding error,
+a missing helper surcharge, a constant factor, or a reducible parity component:
+both phases depend on the other phase and the two-step transfer is positive.
+
+The counterexample also survives one unit of additive work at every internal
+node. If u_i is that recurrence, induction gives
+`t_i <= u_i <= (3t_i-1)/2`, because each profile has coefficient sum at
+least three. Thus adding this positive toll leaves p_eff unchanged.
+
+## 3. The phase-indexed moment gives the right equation
+
+Write z=16^(-p), and let c_A,c_B be positive scale-phase amplitudes in the
+trial function t_i=16^(pi)c_(i mod 2). Its moment equations are
+
+    [c_A,c_B]^T = M(p)[c_A,c_B]^T,
+    M(p) = [[2z^2,z],[4z,32z^2]].
+
+Thus det(I-M)=1-38z^2+64z^4. The positive eigenvector branch at spectral
+radius one is exactly 256^p=lambda, as above. At p=1/2, M has determinant
+zero and trace 17/8, so its spectral radius is exactly 17/8>1. This retains both child
+ranks and the phases reached by each child. Averaging scalar roots discards
+that information. The equation is specific to this finite periodic model;
+no unrestricted spectral transfer theorem is claimed here.
+
+The source's retained local-oscillation error permits this counterexample:
+p changes by 1/2 within the two-interval child window at every large scale,
+so the worst-chain cumulative allowance is of order log x, not o(log x).
+Keeping E in the formula is essential. A sufficient repair to its corollary
+is to establish E=o(log x), including glue and terminal normalization, before
+identifying the asymptotic exponent with the log-Cesaro mean.
+
+## 4. A different proof preserves the finite-catalogue extremal statement
+
+Fix a finite catalogue of positive profiles. Each profile consists of finitely
+many coefficients c_(s,j)>0 and ratios 0<q_(s,j)<1, and has a scalar root
+p_s>=0 with sum_j c_(s,j)q_(s,j)^p_s=1. Choose any profile independently at
+each recursive node, possibly as a function of scale or prior history. Assume
+the recurrence terminates at size <=1, terminal cost is 1, and h=0.
+Let p_min=min p_s, p_max=max p_s and q_min=min q_(s,j).
+
+For every profile, monotonicity in p gives
+
+    sum_j c_(s,j) q_(s,j)^p_min >= 1,
+    sum_j c_(s,j) q_(s,j)^p_max <= 1.
+
+Every terminal descendant of a root x>=1 has size in [q_min,1], apart from
+the root x=1 itself. Hence the leaf bounds are
+
+    y^p_min <= 1 <= q_min^(-p_max)y^p_max.
+
+Propagate these bounds upward through the finite recursion tree. Positivity
+and the two moment inequalities prove, for every node-wise seed choice,
+
+    x^p_min <= T(x) <= q_min^(-p_max)x^p_max.
+
+The finite catalogue has q_max<1, so the tree depth is finite for every x.
+This argument does not assume one common profile at each depth. Fixing a
+minimizing profile gives T(x)=Theta(x^p_min) by the same proof. Thus within
+this zero-glue finite-catalogue recurrence, arbitrary switching cannot improve
+the exponent beyond the best fixed seed. Nonnegative extra work preserves
+the lower bound; an upper bound with glue requires its own estimate.
+
+This validates that limited no-mixture conclusion by a direct comparison;
+it does not validate the unqualified Cesaro equality. Unbounded seed families,
+missing uniform termination/terminal costs, paid representation changes,
+scale-dependent normalization and the actual two-supplier outer transfer
+need separate premises. In particular, no value here is an improved kappa.
+
+## 5. Evidence boundary
+
+The proof covers every scale for the two displayed profiles, and every finite
+recursion tree under Section 4's hypotheses. The executable checks cover
+bounded depths and explicit defect controls. The independent checker and
+REVIEW.md are by a separate GPT-6.1 Sol mathematical worker, not independent
+human or formal-kernel review. The actual selected source, native controller,
+stock/hosting/precision and multiplication transfer are unchanged.
+
+Credit: Philippe Van Velzen's proposed scale-dependent interface and explicit
+error term; standard positive recurrence, companion-matrix and Perron-vector
+arguments. Prepared by Ross Richardson with substantial OpenAI Codex assistance.
+Apache-2.0.

research/scale-dependent-seed-moment-audit/README.md

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+# A scale-dependent seed average can misprice the recurrence
+
+**Scoped mathematical counterexample and corrected comparison theorem; no new
+kappa.** A log-scale average of individual seed roots does not in general
+give the exponent of the complete recursive cost. The missing condition is
+control of the cumulative scale-oscillation error.
+
+This directly tests the changed fixed-seed restriction proposed in
+[Discussion 2278's single-letter lane](https://github.com/CrocSwap/integer-mult-bounds/discussions/2278#discussioncomment-18964674),
+within the mechanism scope of [Discussion 1167](https://github.com/CrocSwap/integer-mult-bounds/discussions/1167).
+The exact inspected predecessor is Philippe Van Velzen's
+`ed21d6245f8ce1a62e21205be7bedc24271dd772`; source paths, Git blob IDs and
+SHA256 hashes are in [SOURCES.json](SOURCES.json). No predecessor code is
+imported or vendored. Replay checks the new mathematical witness locally;
+it does not reauthenticate those remote source bytes.
+
+## Result
+
+In the source model `T(x)=sum_r (n_r/W)T(rx/m)+h(x)`, use m=256, W=1,
+with complete child profiles A=`{16:1,1:2}` and B=`{16:4,1:32}`. Their
+exact scalar roots are 1/4 and 3/4. Alternate A and B on equal log-length
+scale intervals, and initially set all glue/adapters to zero. The averaged
+root is 1/2, but the actual exponent is
+
+    p_eff = log(19+3 sqrt(33))/log(256),    5/8 < p_eff < 2/3.
+
+Both child ranks and all multiplicities are retained. The two-step transfer
+is `[[36,8],[1,2]]`. The positive, phase-indexed moment has spectral radius
+17/8 at the proposed averaged root, so it fails the contraction test there.
+The discrepancy persists with a unit additive toll at every node.
+
+[PROOF.md](PROOF.md) proves the result for every input scale. It also proves
+by power comparison that arbitrary per-node switching in a finite positive
+catalogue cannot beat its best fixed seed under explicit zero-glue,
+bounded-ratio and terminal-cost hypotheses. Thus that limited extremal
+conclusion survives, independently of the invalid unconditional averaging.
+
+This counterexample **does not refute** the predecessor's error-qualified
+Theorem 1: its retained oscillation allowance is order log x here. A claimed
+Cesaro exponent needs that error, and the normalized glue contribution, to be
+o(log x). No verdict is made on the other lanes, Problem X, or amplification
+comparisons. These profiles are abstract positive recurrences; no native
+supplier, physical hosting, precision/storage budget or outer multiplication
+transfer is constructed or admitted.
+
+## Reproduce
+
+From the repository root, using only Python's standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/scale-dependent-seed-moment-audit
+```
+
+The registered commands run with `-I -O -B` and use explicit checks that remain
+active under optimization. They write no package files. Direct portable replay:
+
+```sh
+python3 -I -O -B research/scale-dependent-seed-moment-audit/verify.py
+python3 -I -O -B research/scale-dependent-seed-moment-audit/independent.py
+```
+
+Primary coverage: 240 grid prefixes, 120 exact quadratic-field block powers,
+homogeneous controls, node-wise extremal bounds and eight semantic defect
+controls. The separate checker traverses all rank histories for the displayed
+periodic schedule through index 20, all 1,023 index schedules of length at most
+nine, matrix recurrences through 120 and three semantic controls.
+[REVIEW.md](REVIEW.md) gives its derivation and limits. Finite checks support
+the separate written all-scale proof; they are not a formal-kernel proof.
+
+## Open next questions
+
+For an actual proposed growing-seed family, retain its full child ranks and
+reached scale phases, then bound cumulative oscillation, terminal normalization
+and all glue/adapter costs before assigning an exponent. The first useful
+step is one paid caller and an explicit o(log x) error estimate. This audit
+does not provide that family or justify another seed search. The structured
+question and blocker are in [TASK.json](TASK.json).
+
+## Credit and scope
+
+Philippe Van Velzen's proposed interface and explicit error term motivate the
+audit; the positive-recurrence and companion-matrix comparison arguments are
+standard. Prepared by Ross Richardson with substantial OpenAI Codex GPT-6.1
+Sol assistance. A separate mathematical worker wrote `independent.py` and
+`REVIEW.md` without importing the primary verifier or predecessor implementation.
+This is independent AI derivation, not independent human review. Apache-2.0.
+Selected frontier, frozen inputs and personal configuration are unchanged.