Independent census of C(<=4, <=100) against the four frozen P2 intervals (issue #568)

[LightChainr]

Closes the implementation half of #568. Additive: 13 new files, no existing file touched.

Disclosure, before anything else

scripts/degree4_interval_exclusion.py, scripts/degree4_fixed_point_screen.cpp and
scripts/exact_polynomial_root_certificate.py were never read.

This is nevertheless not a blind replication. The committed summaries
results/pslq-degree4-*/latest.json were read before the replication was written, because
they define the comparison quantities. The code path is disjoint from the primary's; the
priors were not. Recorded in notes/p568-independent-census-20260913.md and in the result
metadata rather than passed off.

What agrees

interval	retention (prim.) mine / primary	near mine / primary	root-bearing mine / primary	closest	residual
jacobsen-2015-eigenvalue	0 / 0	1543 / 1543	0 / 0	agree	agree
mertens-2022-p-med	3 / 3	1548 / 1548	1 / 1	agree	agree
mertens-2022-p-cell	127 / 127	1660 / 1660	15 / 15	agree	agree
yang-zhou-2024-corrected	0 / 0	1543 / 1543	0 / 0	agree	agree

Class sizes agree (157309446881 at d = 4). rho agrees fraction for fraction, and the
screen bound agrees exactly: 20077 / 3000086 / 80000077 / 100065. The closest member
agrees coefficient by coefficient on all four intervals.

Four search paths, not one

tag	structure	scale
local	nested enumeration, leading coefficient solved exactly	1e15
loop	same but the leading coefficient looped — 100× the inner work	1e15
mitm	split {0,1,2} | {3,4}, sorted right half, binary search	1e15
alt	as local, on a different screen lattice	2^50

560 shards over 10 Huawei ARM containers / 144 vCPU, all completed. All four return
identical retained sets, including alt — the retained set does not depend on the
screen lattice, which is the property a correct screen has to have and a wrong one
usually does not.

The check that carries the weight

The primary evaluates the closest member at the endpoints; the midpoint is a different
point, so the residuals must differ, capped at D_P (u−l)/2.

| interval | residual | |P(m)| | gap | cap | holds |
|---|---|---|---|---|---|
| jacobsen | 1.220964e−9 | 1.225831e−9 | 4.867e−12 | 1.284e−11 | yes |
| yang-zhou | 5.930735e−10 | 6.174104e−10 | 2.434e−11 | 6.420e−11 | yes |

Both gaps are inside the cap, and the residuals are 95× and 9.2× the cap respectively, so
this is agreement on a non-trivial quantity rather than two implementations both returning
zero. On mertens-2022-p-med and mertens-2022-p-cell the residual is exactly 0, and the
independent path finds the sign change that makes it 0.

Distinct roots, independently

On mertens-2022-p-cell the 15 root-bearing quartics were checked pairwise by exact
rational polynomial gcd: no pair shares a non-constant factor, so distinct_roots_in_interval = 15 stands on its own rather than on the primary's assertion.

What is deliberately NOT in this PR

The manuscript is not edited. #568 asks that §7's first "Recommended, not done" entry
disappear and that Table 9's scope line stop saying the quartic census is
single-implementation. I am not applying that, because the honest wording depends on the
disclosure above. Suggested text for the owner:

§7, "Recommended, not done", first entry — replace with: "Done, not blindly. PR #NNN
contains an independent re-derivation of the degree ≤ 4, height ≤ 100 census with a
disjoint code path and a different root-decision path, reproducing every cell of §6.1 and
the residual, and additionally checking the 15 surviving quartics of the widest interval
for shared roots. It was written with the committed summaries in hand, so it verifies the
implementation rather than blinding it; a blind replication is still not done."

Table 9 scope line — replace "single implementation" for degree 4 with "one primary
implementation plus one independent re-derivation (PR #NNN, written with the committed
summaries available)".

If the owner prefers the blunt version, applying those two edits is a two-line change on top
of this branch.

Boundaries

No new Monte Carlo, no new p_c, no merge, no docs/STATUS.md edit, no change to any
existing file. Full repository CI has not been run.