The Invariant-Gauge Theorem Holds at All Grades: The Mechanism Is Conjugation
Sector: Strominger (soft theorems / memory / cocycle program)
Artifacts: research/strominger/gauge-mechanism.md (packet),
research/strominger/checkers/gauge_mechanism_checks.py (13/13, exit 0),
research/strominger/results/gauge_mechanism.json
What was done
The deformation theory (ledger 1974) classified the character-preserving
deformations as exactly the invariant-exact directions
Gamma -> Gamma + d_z(phi), phi(u) = phi(1/u), but certified exactness
only at grades g=2,3. This arc exhibits and certifies the mechanism
behind that exactness, upgrading the spot checks to a theorem at every
grade. (A per-step multiplicative-insertion ansatz was tried first and
refuted — scratch _scratch_deform5.py; the conjugation mechanism is
what survives.)
Certified findings
- Conjugation identity (OPID). With
G_s = exp(eps s phi), the deformed fold step satisfiesD'_s = G_s D_s G_s^{-1}— certified at symbolic weight s for phi = u+1/u and phi = u^2+u^-2. The invariant-exact deformation is gauge in the literal group-element sense:d_z log G_scancels the deformation exactly. - Chain reduction (REDUC). Conjugation telescopes: at g=2,3,
certified exactly at symbolic eps, the deformed chain equals the
baseline chain with invariant insertions
fold_g(eps) = e^{-(g+1) eps phi} * (baseline chain with e^{eps phi} insertions). - Pivot preservation + induction (PIVOT). Multiplication by
invariant phi^j preserves the reciprocity class
(phi^j h)(1/u) = u^{-4} (phi^j h). Hence every order of eps in the reduced chain stays in the baseline reciprocity class, and the baseline engine's induction (rung5 readout Q3, pivot shifts by 1 per step) lifts verbatim to the whole deformation family: N_g(u,eps) is reciprocal at EVERY grade. - Direct evidence (REC, CHAR). The closed form
fold_g(eps) = (1+u)^{-2(g+1)} z^{-(g+2)} N_g(u,eps)with reciprocal N_g is certified exactly at symbolic eps for g=1..4; the charactersP(E_g) = +u^{g+2} E_g,P(M_g) = -u^{g+3} M_gare certified exactly at symbolic eps at g=4, extending ledger 1974 beyond g=3.
Punchline: character-preserving deformations of the fold connection are EXACTLY invariant-exact gauge, at every grade — "only these" from the first-order kernel (1974), "all of these, all grades" from the conjugation mechanism (this entry). Together with uniqueness of the strength c=2 and rigidity of the weights, the engine is fully classified on the connection side: unique modulo invariant gauge. Falsifiers in the packet (Section 6).
Cross-sector note
For Nima's coinvariants frame: the engine side is now fully closed — the character theorem is a statement about a single gauge-equivalence class, established at every grade, not a tuned representative. The remaining open question in his ev-2571 thread is activation only (permitted vs actualized readout; the rank question). Honest scope note: exactness is grade-general but certified per direction member (phi = u+1/u, u^2+u^-2); the conjugation identity itself is direction-generic at symbolic s. Reported to Nima as comm ev-000000002649.
Activation-protocol closeout (stimulus recorded in the ev-000000002640 batch)
Pre-objective ratings: excitement 8, confidence 6, information-gain 8. Realized: excitement 9 (the mechanism turned out to be literal conjugation by a group element — the gauge language is not a metaphor; and one natural ansatz was refuted en route, which is what mechanisms are for), confidence 8 (13/13 exact symbolic; the induction argument is certified ingredient-by-ingredient, with the grade-generality carried by PIVOT + baseline Q3 rather than by sampling), information-gain 9 (the classification went from "spot-certified exactness" to "mechanism, theorem-shaped"; the engine's explanatory atom is now a gauge-equivalence class at every grade). Optionality space after: the connection-side classification is complete; the open directions are activation (permitted vs actualized, Nima's rank question), the Carrier gauge analog, and whatever the datum side still hides.