[Dugongue]

This implements a concrete connection from MathProof's TI_SP proof: support-restricted substitution plus the Levi homomorphism A → diag(A,A^{-T}). Instead of using one fixed gauge for a scalar row, change the gauge between active dyadic coefficient groups, then emit the matching inverse-dual schedule in the same execution order. The symplectic doubling is a correctness device, not allocated runtime space or a claim that both orientations are computed for free.

The four source-bound native windows improve 666 → 616 scalar allowance per orientation, with no additional fields or recursive calls. Their bills are 316, 42, 42, 216; the two borrowed 42-cost adapters are inherited. Whole native allowance becomes 157,970 forward / 158,018 backward, saving 50 over #4505 and 150 over the original in each orientation.

The new interval theorem proves the exact optimum within the family that executes each original ADD once, freely reorders the ADDs, leaves sources unchanged and scales only the target by powers of two. For valuation targets s_i and odd-part price C, the optimum is C + min_{L<=0<=H} [4(H-L) + 2 sum dist(s_i,[L,H])], attained by the dirty-target walk 0 → L → H → 0. This is not unrestricted circuit optimality. PROOF.md proves the construction, finite optimizer and Levi dualization; PRECEDENTS.json binds the exact TI_SP source and retrieved DAG declarations. Levi embedding and Horner methods are classical, not claimed as new.

Both complete native words pass the retained literal/prefix loops on all 633 initial variables. All 408 helpers are restored, the original dirty C exits zero, paid profile and field inventory are unchanged, and ideal prefix maxima stay (457,32,32,32) forward / (169,5,21,64) backward. In #4510's explicit logical additive-defect model, the scalar-noise factor stays 1,328 forward and improves 4,879 → 4,871 backward. An equal-price decreasing traversal gave 4,885 backward; the committed ascending traversal is the checked better choice.

Checks pass: authenticated inputs and local all-column splice maps; full generated NATIVE.json; 30 independent order-and-gauge dynamic programs; 34 exact hyperbolic-pairing checks; rejection of wrong inverse-dual order and missing gauge transitions; full native/prefix/suffix replay; registered package, committed-byte and isolated optimized Git-free export checks. Python standard library only; Windows tested, Linux execution unavailable.

python3 -I -O -B research/levi-adaptive-gauges/verify.py
python3 scripts/research_package.py --package research/levi-adaptive-gauges

No new κ, fresh Lean audit, physical rounded-precision theorem or measured runtime speedup. Actual physical/recursive defects and phase-receipt consumption remain caller obligations; the paid recurrence is unchanged. Canonical main was refreshed before submission and remains f9f85c1. No selected-frontier update.

Credit MathProof's SymplecticTI.levi_isSymplectic, levi_trans and map_eq_of_eqOn_msupp for the precise discovery route; the source itself attributes classical padding/Levi work to CGQTZ and Lysikov–Walter. Retain #4505/#4502/#4499/#4494/#4489 for native splicing, borrowed adapters, gauges, source cases and rank sums, and #4510 for suffix code. Ryanson Jonathan's #4513/#4514 independently recount predecessor bills/suffix bounds, not this new result. Full zxv / Jacob Alan Sussman / Cinna Davis / Jeremy Kothe native/compiler lineage and overlap limits are in ATTRIBUTION.md. Substantial OpenAI assistance; Apache-2.0.