Unconditional endomorphism-ring equivalence

Setting

• CITED Supersingular elliptic curves over $\mathbb F_{p^2}$ have endomorphism algebras isomorphic to the quaternion algebra $B_{p,\infty}$ ramified exactly at $p$ and $\infty$ (Deuring 1941).
• Problem A: given supersingular $E_1,E_2/\mathbb F_{p^2}$, find an isogeny $E_1\to E_2$ of smooth degree.
• Problem B: given supersingular $E/\mathbb F_{p^2}$, compute $\operatorname{End}(E)$ as an explicit maximal order.

Targets

1. Prove that Problems A and B are equivalent without assuming GRH.
2. Separately quantify the concrete security loss of the reductions as a function of $p$.

First deliverable

Produce a GRH usage map for the existing equivalence proof: identify every GRH-dependent lemma, the exact effective bound it needs, and the strongest relevant unconditional substitute. Mark reconstructed entries UNVERIFIED when the source text cannot be checked.

Success conditions

• Task 1 succeeds only with an unconditional equivalence proof or a precise necessity result for GRH.
• Task 2 succeeds only with an experimentally validated concrete loss function.
• A source-checked GRH usage map is a useful partial result, not a solution to Task 1.

State in five lines

PROVED P3.1 is partial, not resolved. PROVED A003 gives a repo-internal unconditional replacement for D2; D1, D3, and D4 remain. PROVED A002, A004, A005, and A006 are dead with precise mathematical post-mortems. EMPIRICAL: p=11, ell=3, 30 independent seeds The right-order-aware dual two-step Deuring fixture passed 30/30 trials. EMPIRICAL: p in {11,23,47,59,71}, ell=3, n=5 Five exhaustive toy timings have stored residuals, but they do not identify security-bit loss.

What is established

• PROVED GRH_USAGE_MAP.md classifies the original proof's four direct leaves D1--D4 and all inherited uses.
• PROVED A003 replaces Proposition 3.8 by direct rank-four residue-lattice sampling using effective quaternary theta bounds, explicit primes in progressions, and fixed-rank CVP.
• EMPIRICAL: p<=31, ell in {3,5}, q<=3000 A003's 22-case refutation grid had no reciprocity violation; scaled prime density ranged from 0.832 to 1.815.
• EMPIRICAL: p=11, ell=3, 30 independent seeds Every ideal, right-order, dual-product, and terminal-curve check matched; all four embedded neighbor orders had discriminant 121, every product was $I\bar I=3O$, and every dual quotient returned the source class, with Wilson 95% interval approximately [0.886,1].
• EMPIRICAL: p in {11,23,47,59,71}, ell=3, n=5 The exhaustive right-order-aware two-step cost exponent is 1.6796, with classical 95% interval [1.2366,2.1226] and R-squared 0.9798.

What is ruled out

• PROVED A002: Brandt mixing does not control the deterministic binary-subform pushforward needed by average-form theorems.
• PROVED A004: an integral orthogonal 2+2 split cannot hide the ramified prime p from every modulus/discriminant/index.
• PROVED A005: ellipsoid rejection for one fixed quaternary target loses essentially a factor n and is exponential in log n.
• PROVED A006: the flexible dictionary removes D1 from conversion, but not the small-discriminant dependency inside the old norm-equation solver.

Active thread

A003 is promising and removes D2 only. The main theory target is a constructive fixed-target quaternary solver on arbitrary maximal-order lattices. The implementation thread is SG-03f, non-backtracking dictionary propagation across a second ideal.

Next action

For p=11 and ell=3, represent a non-dual norm-3 ideal in the first embedded right order, propagate its action to the intermediate curve, and identify the terminal right order without an exhaustive curve-key lookup.

Invariants - do not violate

• PROVED Keep unrestricted Isogeny separate from prescribed/smooth ell-IsogenyPath.
• PROVED Treat A003 as a repo-internal proof candidate, not a published resolution of P3.1.
• PROVED D1, D3, and D4 remain; do not report the full equivalence as unconditional.
• PROVED The timing fit measures exhaustive toy enumeration and cannot be used as a security-bit loss.
• Preserve the current tuple-based PrimePolynomialField API in lib/finite_fields.py.

Files that matter

GRH_USAGE_MAP.md; attempts/A003-direct-quaternary-prime-sampler.md; TIGHTNESS_STATUS.md; DEURING_ROUNDTRIP_SPEC.md; REDUCTION_COST_SPEC.md; lib/quaternion.py; code/toy_deuring_roundtrip.py; code/fit_roundtrip_cost.py; data/fit_roundtrip_cost_ell3_p11-71_20260711.csv.

What I would tell my replacement

The modern flexible dictionary is not the missing smoothness theorem. It removes the need for a special curve at conversion time, but the old ideal-smoothing core still depends on a D1-small quadratic suborder and on D3+D4. Attack the constructive fixed-target quaternary equation or complete the right-order round-trip fixture; do not return to least-prime bounds alone.