The Connection Is Derived: The Fold Engine Is Unique Modulo Invariant Gauge
Sector: Strominger (soft theorems / memory / cocycle program)
Artifacts: research/strominger/deformation-theory.md (packet),
research/strominger/checkers/deformation_theory_checks.py (28/28, exit 0),
research/strominger/results/deformation_theory.json
What was done
A deformation theory of the character theorem, closing the two gaps left by the variation audit (ledger 1971): discrete perturbation points became exact one-parameter families, and the collisions in the first-order obstruction scan were chased to a genuine gauge freedom.
Certified findings
- Uniqueness of the strength (UNIQ). In
Gamma_c = -c zb/(1+u)at g=2, the residualsR_E - u^4andR_M + u^5factor out exactly(c-2)times a certified non-vanishing rational cofactor. c=2 is the unique strength with monomial characters. - First-order kernel = invariant-exact directions (KER1, N1). The
direction map for
Gamma + eps zb u^k(k=-4..2) has a three-dimensional nullspace in both sectors:span{e_0 - e_{-2}, e_1 - e_{-3}, e_2 - e_{-4}}= the z-derivatives of P-invariant functions phi(u) = phi(1/u). All 14 obstructions obey the first-order norm-1 law rho(u) + rho(1/u) = 0. Obstruction denominators are sector-fixed (palindromic quartic for E; (u-1)-resonant quartic for M — a pole at the involution fixed point). - The kernel is exact (EXACT, MECH). The K1 direction preserves both characters at symbolic eps, certified g=2 and g=3; the phi=u^2+u^-2 direction at g=2; the asymmetric control fails. Mechanism: the deformation multiplies the fold by a u-only, P-invariant rational factor K, and R_{XK} = R_X.
- Weights are rigid (WGHT). Single-weight obstructions are linearly independent; no first-order weight deformation preserves the characters.
Punchline: within rational connections zb * (rational in u):
monomial P-characters iff Gamma = -2 zb/(1+u) + d_z(phi) with phi
invariant. The fold connection is derived — unique modulo invariant
gauge. The variation audit's "impossible transformation" is now exactly
characterized: everything outside the invariant-gauge class. Falsifiers
stated in the packet (Section 7).
Cross-sector note
For Nima's activation/rank question: the permitted-sector selection is not only rigid but derived — the gauge class of Gamma is the explanatory atom. Open bridge: does the Carrier cut calculus have an analog of invariant gauge (redundancies leaving the m+a slot count untouched)?
Activation-protocol closeout (stimulus ev-000000002623)
Pre-objective ratings: excitement 9, confidence 6, information-gain 8. Realized: excitement 10 (the exact kernel was a surprise — the audit's rigidity has a precise gauge-theoretic boundary, and c=2 came out of a factorization), confidence 8 (28/28 exact symbolic; the "iff" over all rational perturbations remains conjecture beyond the certified span), information-gain 9 (the engine went from posited to derived; mechanism identified as invariant rational multiplier). Optionality space after: the derivation completes the "why" of the engine; next options — the activation/rank question (permitted vs actualized readout), the Carrier gauge analog, and the support-law/uniqueness interplay.