Exact domain separation for phase 2

The 81 configurations below can search candidate positions (D,r,z) that are disjoint from the completed phase 1 production campaign and from one another. This is a mathematical statement about these finite domains, conditional on exact root checks, correct interval endpoints, and nonoverlapping completed ledger reservations. It is not an exhaustive height bound, a proof of a solution, or a discovery probability. Interrupted partial work can be repeated on recovery.

python3 verify_domain.py verifies the numerical inequalities with rational arithmetic, reruns the PARI field/unit certificate, and records evidence in runs/domain-certificate.json. An optional --config FILE checks the complete context table against the fixed geometry. The derivation has independent mathematical review but is not a machine-checked proof of the C and Python implementation.

1. The finite domain and its required guard

Write alpha³=114, S=10^18, p=4848807585839879338, and q=23510935004498358840. For ell∈{1,5,25}, set

a0 = (−4b−16c) mod ell, with 0≤a0<ell,
u  = −p b−q c−a0 S,
base = a0+ell floor((2u+ell S)/(2ell S)),
a = base+ell t,
gamma = a+b alpha+c alpha².

The integer formula rounds to the closest class-lattice point, breaking exact half ties toward positive infinity. Both b and c include every integer in [-H,H]. The three shapes are exactly:

Let D0=floor(10^19/54)=185185185185185185. Each shape and ell is paired with all three lower-exclusive, upper-inclusive norm shells (D0,2D0], (2D0,4D0], (4D0,8D0], and ratio bands (0,64], (64,256], (256,4096] for |z|/D. This gives 3×3×3×3=81 contexts. The ratio variable is |z|/D, not the integer quotient in z=r+Dq.

The scheduler and learning model must select identifiers from this fixed table. They must not invent radii, offsets, ell values, shell boundaries, or ratio endpoints. validate_contexts in the verifier checks all 81 geometry tuples, rejects duplicates and out-of-table values, and can be called before launch and after loading saved configuration. Reservations and permutation/index bijections remain separate controller obligations. Geometrically identical contexts with different seeds do not become disjoint domains.

2. Positive real embedding and shape separation

Cubing the rational endpoints proves

|alpha−p/S|<1/S,   |alpha²−q/S|<1/S.

Thus the positive real embedding s of gamma obeys

ell(t−1/2)−epsilon < s < ell(t+1/2)+epsilon,
epsilon=2H/S.

The smallest lower bound is positive. Consequently Norm(gamma)=s|sigma_complex(gamma)|²>0: this nonzero field element has positive norm, and neither it nor its negative can appear twice through a sign rule. The three real-embedding intervals for the same ell are strictly disjoint, despite their different H. Their combined largest/smallest possible embedding ratio is less than 1093.

The certified field has integral basis (1,alpha,alpha²), class group cyclic of order 3 generated by J=(5,alpha−4), and J²=(25,alpha−4). Its unit group is generated by −1 and

beta=4133238949+852423792 alpha+175800705 alpha²,
beta>4133238949, Norm(beta)=1.

The fresh PARI certificate verifies that beta is fundamental, rather than only a unit. Two generators of the same principal ideal have absolute real-embedding ratio beta^m for integer m. The new range is much shorter than one unit period: equal ideals within a class force m=0, and positive real embeddings then force the generators themselves to be equal. Identical coefficients also force the same b,c,t, so different shapes cannot hide another representation of the same ideal.

3. Root ideal ownership across the three classes

Define

N=a³+114b³+12996c³−342abc=ell D,
A=a²−114bc, B=114c²−ab, C=b²−ac.

Every sampled gamma belongs to J^j, where ell=5^j. The root extractor must require gcd(C,D)=1 and verify r³≡114 (mod D) for r=B/C mod D.

For a surviving candidate in these classes, in fact gcd(C,N)=1. The only additional possible bad prime is 5. If j>0 and 5|C, lattice membership gives a=b−c (mod 5) and b²−bc+c²=0 (mod 5). This binary quadratic has no nonzero zero over F5; hence a,b,c all vanish modulo 5 and 125|N. If 5∤D, that contradicts v5(N)=j≤2; if 5|D, the existing gcd check already excludes it. For j=0 the two gcd statements coincide.

The exact polynomial identities

B³−114C³ = N(114c³−b³),
B²−AC = cN,
aC+bB+cA = 0

then prove that R=B/C mod N is a cubic root modulo N and that gamma(R)=0 mod N. Inclusion and equal index give (gamma)=I(N,R), where I(m,v)=(m,alpha−v).

Modulo 5, X³−114=(X−4)(X²+4X+1) and the quadratic is irreducible. The simple root 4 lifts uniquely to every 5-power. Extending that prime-power root and applying CRT away from 5 gives

(gamma)=J^j I(D,r),
[I(D,r)]=[J]^(−j).

The three j values therefore own different root-ideal classes: different ell cannot emit the same (D,r). Within an ell, the unit argument above proves that two new generators cannot emit the same (D,r) unless their coefficients are identical.

The complementary completeness lemma is also valid: for an actual generator (gamma)=I(D,r)J^j, the quotient O_K/(gamma) is cyclic of order N. If a prime divided C and N, the identities B³−114C³=N(114c³−b³) and A²−114BC=Na would force A,B,C all zero modulo that prime. The multiplication matrix's adjugate would vanish, forcing rank at most one and a cokernel of dimension at least two, contradicting cyclicity. This proves invertibility for the certified ideal-to-generator construction, including at ramified primes. It does not make arbitrary sampled elements complete representatives of all root ideals. The explicit weak-converse counterexample is documented in ../unit_shape_proposal.md.

4. Separation from completed phase 1

It suffices to compare generators of the same ell and the same D; other ell are excluded by class ownership. Phase 2 always has D>D0. Let n=ell D and use n_min=ell(D0+1) in lower bounds.

Phase 1's box radii are (50000,85499,146201) for ell (1,5,25). Both its absolute real and complex embeddings are at most B=31H+ell−1, including the asymmetric a-lattice boundary. Thus its absolute real embedding satisfies s_box≥n/B² and s_box≤B. If L_new and U_new are the smallest and largest bounds from section 2, exact rational checks give

1 < n_min/(B² U_new) ≤ s_box/s_new ≤ B/L_new < beta.

The lower ratios for ell 1,5,25 are respectively greater than approximately 9.4097, 3.2180, and 1.1005. The narrowest margin is positive with the stated constants. The largest upper ratio is less than 206667, far below beta. There is no integer power of beta between these endpoints.

Phase 1's plane has radius 20,000,000 and five nearest-lattice offsets. It satisfies s_plane<3ell and w_plane<10^9, hence s_plane>n/10^18. Exact checks give

1 < L_new/(3ell) < s_new/s_plane < U_new·10^18/n_min < beta.

The last upper ratio is below 44235. The new regions therefore lie strictly between the old plane and box in absolute real embedding and cannot differ from either by a unit. The proof applies to all positions in those phase 1 domains, which is stronger than comparison with only their completed sampled prefixes. It does not apply to unrelated campaigns or calibration experiments.

5. Shells, ratio bands and ledger composition

The three D shells have distinct interiors and allocate each shared endpoint only to the lower shell. The ratio bands do the same for |z|/D; the worker must implement this with exact integer comparisons. For a fixed D,r, every integer z has a unique quotient in z=r+Dq with 0≤r<D. Therefore repeated root curves in distinct ratio bands correspond to different z positions. A reported curve-check count must still not be described as a count of distinct root curves across bands.

Combining these endpoint partitions with root-ideal ownership, unit separation, unique row/offset indexing, and disjoint completed reservations proves no repeated (D,r,z) candidate positions within completed phase 2 work or between completed phase 1 and phase 2 work. Worker retries can repeat unfinished partial work; a mathematical domain proof cannot prevent that operational duplication.

6. Arithmetic limits

The verifier bounds every proposed a by 170,159,244 in absolute value; |b|,|c|≤6,000,000. A sum-of-absolute-terms bound for the norm polynomial is below 2^84, so signed 128-bit norm and difference arithmetic have ample room. 114|bc|≤4,104,000,000,000,000<2^63, allowing an exact integer square root on the derivative threshold in a 64-bit implementation. All rounding numerators, adjoint coefficients, row counts, and possible individual coefficient-index counts also fit their documented integer types.

D≤8D0=1481481481481481480<INT64_MAX. The norm ell D can exceed uint64 and must remain in signed 128-bit storage until division by ell. |z|≤4096D fits signed 128 bits, but z cubed and the discriminant do not: final square/cube arithmetic must continue to use PARI arbitrary-precision integers. This section proves bounds for the formulas, not that every C expression has the correct type promotion; native validation remains required.

7. What the learning reward can honestly mean

A reward of sum(D0/D × F_band)/elapsed_time is a density-weighted throughput score. Under the additional model that sampled roots are exchangeable with roots in the asymptotic counting heuristic, a root near D carries mass proportional to 1/D, and the geometric ratio-band factor F_band apportions that heuristic mass. This can compare computational effort across shells and bands more sensibly than raw curves per second.

Exchangeability is unproved for these structured norm families, especially after selecting by unit shape and ideal class. The reward is therefore not a measured success probability, and learning a larger value does not demonstrate enrichment for solution-bearing curves. Zero hits provide almost no supervised signal for discovery. Modulus-filter survivors, exact-square near misses and large candidate counts are also not proven predictors of a hit.

Use actual verified checked-curve weights, complete-worker wall time including overhead, held-out execution measurements, and continued exploration of the finite context table. Treat adaptation as cost optimization until independent evidence establishes a predictive advantage. Failed exact tests exclude the tested positions; the model must not turn statistical pessimism into a mathematical exclusion of untested regions.