Checkpoint: class-structure-2026-10-08. Owner: delegated class_structure. Read-only repository base f4ef0715bbe2f4890b3dcea034757ef0c823053a. PR279 inputs pinned at 3f136908426167ff97888a6ae8b89087223509e6. Write scope: task-3/routes/class-structure only. No Git changes or external compute. Applied mathbox research-program, proof-audit, literature-check and computation-audit. Repository mission/results/runs and C/reference code read.
Outcome
A useful candidate enhancement of the existing lower-bound sieve survives an exact small-range test. It is an application of classical genus theory, not a novel theorem or a completed classification. Whether the gain pays for factoring and character evaluations at H=1500 is still unmeasured.
For all 4,055 odd negative fundamental discriminants with |D| <= 20,000, counting only reduced forms with a <= 10, the genus bound improves even the divisibility-rounded total-count bound in 1,918 cases. It rejects 330 extra discriminants for H=20, one extra for H=50, and none extra for H=100. These are threshold comparisons, not runtime improvements.
At D=-15015, fourteen such forms give a plain bound 14, or 16 after genus divisibility rounding, while the genus bound gives 64. The full exact count is 96. The factor-four improvement compares bounds, not speed.
Exact target and argument
Let D=-n be odd, negative, fundamental, and n the product of t distinct odd primes, with n=3 mod 4. Let S be any set of distinct canonical reduced primitive positive-definite forms of discriminant D. For each prime p|n, evaluate (m/p) at a represented integer m coprime to n. The vector of these signs is the genus signature. Write N_s for the number of forms in S with signature s, and g=2^(t-1).
Classical genus theory supplies a surjective homomorphism from the proper class group to the sign vectors of product +1, with kernel the square subgroup. Consequently each of its g fibers has exactly h(D)/g elements. Distinct canonical forms represent distinct proper classes. Thus
h(D) >= g max_s N_s >= g ceil(|S|/g) >= |S|.
The middle inequality is the pigeonhole principle. A correct rejection rule for h(D)<=H is max_s N_s > floor(H/g). This conclusion is unconditional given the classical genus-theory input; no zero-free hypothesis is used. The finite implementation currently supports only odd fundamental D.
More generally, for any surjective homomorphism of finite groups G -> Q, the same bound |G| >= |Q| max_q |S intersect fiber(q)| holds. The proof is just equal coset cardinalities. This generality is classical and is not claimed as research novelty.
The existing cn.c filter sums partial forms without this partition. The proposed adaptation would retain a small counter per genus for survivors, after cheap filters, and stop as soon as any counter crosses floor(H/g). Doing this before streaming could lose the existing periodicity advantage. No production implementation or speed claim has been made.
Falsification and limits
Every discriminant in the stated finite domain passed full genus uniformity, the product-sign convention, g distinct signatures, exact h comparison with the pinned PR279 reference counter, and all three bound inequalities. The two form counters share the mathematical reduced-form definition, but use different boundary-loop conventions; this is not an independent PARI implementation. All decisions use Python exact integers.
The tempting shortcut of multiplying the total count by g is false already at D=-15: h=2, g=2, so that shortcut says 4<=2. The code checks this decoy. Ramified leading coefficients cannot be fed into a character as though they were coprime; the implementation finds a coprime represented value instead. Its finite search for that value raises an error after 101 trials; it never silently omits a form. The canonical reduced-form duplicate check is explicit.
Genus checks cannot prove completeness of an enumerated list. At D=-39 the four reduced forms are (1,1,10), (2,-1,5), (2,1,5), (3,3,4). The first and last have signature (+,+), the middle two (-,-). Deleting one form from each genus leaves a false count h=2 that passes divisibility, the number of genera, and genus uniformity. Thus this invariant cannot fill PR279's missing all-discriminant streamed-filter audit. This is an exact counterexample to that proposed audit criterion, not to PR279's result.
Sources and overlap
Cremona and Sutherland, Computing the endomorphism ring of an elliptic curve over a number field, arXiv:2301.11169v4, section 5, checked 2026-10-08: https://arxiv.org/html/2301.11169v4#S5 Their 4,115,897 fundamental discriminants with h<=1000 and the largest |D|=227932027 match PR279's totals. Their all-orders list has 6,450,424 discriminants and assumes GRH for completeness. Accordingly PR279 is not the first even-class-number classification above 100.
Genus theory is a classical external input. Primary-source retrieval for its exact formulation was unsuccessful in this session (the attempted Conrad PDF was inaccessible, and searches returned mostly irrelevant results). The universal genus-to-filter implication is therefore marked conditional on that named external theorem pending exact-source verification in this audit, despite the 4,055 finite checks. No novelty claim is supported by this bounded search. A source-checking reviewer should use the principal genus theorem for imaginary quadratic fields and the odd fundamental discriminant genus-character formula; verify surjectivity, not only the character map.
Reproduction and independent audit
From zeta-research:
.venv/bin/python ../routes/class-structure/probe.py
Python 3.13.14, one process, hard CPU cap 20 seconds, measured about 2 seconds. No numerical libraries or network required. Output in probe.json. The exact row hash is 52cc9a4beb6fff3a7a6d7a1838216c007c1c17035abbc1bf13a86df849d9071f. reference_pinned.py is the unmodified classno.py from the pinned PR279 SHA.
Independent audit path: derive the equal-coset bound from the stated class group map; check signatures using a second coprime represented integer; recompute full h using PARI qfbclassno or another reduced-form enumerator; verify the D=-39 deletion decoy; then benchmark the survivor-stage filter against cn.c on disjoint windows without changing the recorded master data. Do not interpret a matched hash as proof of a universal statement.
Status: promising implementation experiment; universal application awaits exact-source review, H=1500 relevance and net speed remain open. No new classification or genuinely new mathematical discovery has been established.