research: prove growing lower bound for branching predictive classes

[LightChainr]

Purpose

Advance the cut-network / predictive-state paper from a single finite no-go witness to a genuine growing-family lower bound.

This PR now contains two logical steps:

1. the PR Exact: full survival laws do not close the topology state (#429) #435 N16 witness implies that complete unbranched survival is not sufficient for delayed branching;
2. a new parallel-gadget construction amplifies that witness to k+1 exact branching-predictive classes inside one identical complete-survival class on 8k future vertices.

No Monte Carlo, new descriptor ladder, continuum claim, or production change is introduced.

New theorem: parallel-gadget lower bound

The two PR #435 gadgets have the same safe-subset polynomial

S(z)=1+7z+18z^2+20z^3+8z^4

and the same first successor-hazard moment sum x=13, but distinct second moments

q_A=29,
q_B=25.

Use PR #491 to view them as two-terminal cut-network gadgets. Put k future-vertex-disjoint copies in parallel and let a of them be type A.

Because parallel safety means every gadget is safe, the full fixed-cardinality safe-subset polynomial is exactly

S_k(z)=S(z)^k,

independent of a. Thus all a=0,...,k systems have exactly the same complete unbranched survival law.

For the same one-shared-update / two-clone delayed fork used in PR #435, the exact probability is

F_{k,a}
 = [343 k^3 - 182 k^2 + 25 k + 4 a]
   / [8 k (8k-1)^2],

so

F_{k,a+1}-F_{k,a}
 = 1/[2k(8k-1)^2] > 0.

Therefore one complete-survival equivalence class splits into at least k+1 exact branching-predictive classes on 8k future random vertices. Equivalently, this explicit family has a linear lower bound

predictive classes >= n_future/8 + 1.

This removes the need to infer unbounded predictive complexity from the finite N16/N17 census.

Embedded scope

The construction stays inside the PR #491 finite embedded-graph/rank-one continuation category: place the planar cut-network gadgets in disjoint strips of a cylinder sharing only deterministic terminal boundaries, then glue the cut boundaries back to a torus. It does not claim every family member is a nearest-neighbour square-site HNF quotient.

Files

• notes/p429-survival-law-no-go-theorem-20260901.md — finite no-go synthesis;
• notes/p429-parallel-gadget-lower-bound-20260901.md — growing-family theorem and proof;
• scripts/p429_parallel_gadget_lower_bound.py — exact Fraction/integer verifier;
• tests/test_p429_parallel_gadget_lower_bound.py — base witness, convolution, closed-form and strict-splitting tests.

Claim boundary

This PR proves unbounded state cardinality / predictive-equivalence-class growth for an explicit embedded family. It does not prove:

• Euclidean state-dimension growth;
• continuum field-count lower bounds;
• minimality of the full PR Prove the rank-one cut-network theorem and explain the real trigger witnesses #491 cut network;
• the same lower bound for every HNF or nearest-neighbour square-site torus;
• a population-level [P1 analysis/observable] Projective essential-birth spectroscopy: mark the first ambient-H1 birth by its winding line #334 effect;
• a CFT/Jordan/operator identity.

One real-valued scalar can encode arbitrarily many discrete labels, so the theorem is deliberately not stated as a scalar-dimension no-go.

Next PRIMARY and stop rule

The next useful theorem is now minimality up to network equivalence: identify a canonical quotient of the cut network preserving all unmarked shared-prefix/fork experiments, or give an exact counterexample to a natural quotient such as two-terminal reliability plus bounded terminal-local data.

Do not create larger parallel products merely to increase the k+1 count; the unbounded lower bound is already exact. Further work must reduce the constructive state or prove a stronger no-quotient statement.

No main write, auto-merge, Issue lifecycle change, or production authorization.