[petalas]

Summary

Scoped proof with exact finite corroboration; no new kappa. Joint execution cannot reduce the sum of the standalone transition charges for the Gaussian-dyadic blocks 2^a (1+epsilon*i)^b, with integer a, b in {0,1}, and epsilon in {-1,1}, even with arbitrary cross-helper mixing and restored arbitrary dirty workspace.

minimum scales = sum |2a+b|
minimum ADDs   = 2 sum b
minimum total = their sum, attained simultaneously

(1+i), (1+i)^-1: exactly 4 ADDs + 2 scales = 6

This answers Open next question 2 of Sagar Thakrar / thakrarsagar's #1566. Its standalone transition bounds, temporal-gauge setting and Hanson Sun / bbMisk's #1490 common-rounding lineage retain their credit. The new step is a joint lower bound using classical lattice-index distance and row support. No predecessor is audited or reverified. The proof, construction and verifier were produced by Nick Petalas.

Evidence

• Before, Scoped negative result: common rounding for temporal Gaussian gauges #1566 excluded joint transitions from its theorem because opposite determinant valuations might cancel. The reciprocal pair has total determinant one. After, the lattice-index distance still forces two individual scales and row support forces four ADDs; explicit words attain both. The written argument covers every finite helper count, every integer exponent and any finite restored dirty workspace.
• The standalone verifier enumerates 1,011 attaining endpoints with literal rational matrices and inverse restoration, 42 finite lattice quotients, 216 lattice triangles, all 4,681 words through length four over the eight-gate small alphabet, and 14 rejecting controls. The generated receipt binds all other package files by SHA256. Finite checks corroborate the universal written proof.
• Warning-strict normal and optimized CPython 3.12.15 pass locally on macOS 27.0.1 arm64. Registered commands took 0.53 and 0.36 seconds locally, each below its 120-second timeout. The complete registered runner, indentation check and committed-byte/source-history preflight pass. Relocated read-only Git-free replay and receipt/source/CLI corruption controls pass. No assertion is required for acceptance.

From the worktree root, use these exact local reproduction commands:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/joint-gaussian-transition-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/joint-gaussian-transition-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/joint-gaussian-transition-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/joint-gaussian-transition-floor --check-source-history

Any CPython 3.12 interpreter can replace the absolute path. Standard library only. Expected markers are PASS_JOINT_GAUSSIAN_TRANSITION_FLOOR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Default replay writes nothing. --emit-result prints the generated receipt to stdout for external storage.

Limitations and unrun obligations

The alphabet counts signed unit ADDs, individual doublings and halvings. Coordinate permutations and negations are granted free. The required endpoint is the pure block-diagonal transition with identity on every dirty workspace coordinate. A separately required return has the same cost; a round trip with no intermediate requirement can be empty.

Fusing transitions with original operations, other coefficient families, singular clean workspace, COPY/clear, nonlinear gates and larger native primitives are outside scope. Routing, precision, ownership, complete scalar-word optimization, paid profiles, compiler/stock/outer admission and a complete multiplication theorem are not established. No Lean, Docker or selected cold replay was run. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

Open next questions

Can a transition fused with one actual signed SCALE or ADD beat the complete corresponding real-gauge invoice? Start with its literal dirty-input endpoint and an attaining word. A source-bound fused boundary and complete caller invoice remain missing. Further pure joint-transition searches cannot evade the proved charge in this alphabet.

Related open work

• Save six more scalar units after P104 sign normalization (no new κ) #5969 simplifies ordinary P104 ADDs after sign normalization. This proves an isolated Gaussian transition floor and does not compute or compare that word's scalar saving.
• Save twenty P104 scalar units and two events through dirty-output sharing (no new κ) #5999 shares a P104 linear form through a dirty destination. Its changed non-block-diagonal endpoint is outside this theorem; no cumulative saving or competing transition count is claimed.
• Prove optimal four-addition, one-halving Pauli pairing normalization (no κ) #6003 proves an elementary normalization tariff for an anticommuting-Pauli three-bank pairing. Its operator-valued additions and non-block-diagonal endpoint differ from these scalar Gaussian transitions; there is no shared computed endpoint cost.
• Prove an integer-shear target and its inverse need eight calls: no cross-role sharing (no new κ) #6011 proves a target/inverse minimum in the formal integer-shear group-label model. Its paid group-label factors differ from the scalar ADD/scale alphabet here; its inverse-pair count is not a competing value for this Gaussian endpoint.
• The live search found no open PR studying the same pure joint Gaussian transition endpoint. Merged Research: joint L253/L254 temporal schedule on PR1263 word #1732 optimizes temporal gauge choices with separately charged transitions; it does not lower their joint elementary implementation. Merged Reduce Gaussian phase payload work by 25–50 percent #5370 prices Gaussian unit phases on tapes, a different primitive and cost model.

CI notes

Hosted CI passed at exact head 7aed721cfdbf26ed653b9667a413ecd2689b83db: plan and replay (research/joint-gaussian-transition-floor), in run 40584835398. The completed run has conclusion success and its head SHA matches the PR head. Local registered normal/optimized checks and committed-byte/source-history checks also pass. Hosted replay certifies the registered bounded checks; the universal quantifiers rest on the written proof.

Merge Danger

Door: two-way. Blast radius: package. Seven new files under research/joint-gaussian-transition-floor/; no shared code, predecessor package or selected-frontier file changes.