[petalas]

Summary

Scoped analytic proof with exact finite corroboration. The broad Gaussian inverse at a phase-cell edge has sharp order Theta_alpha(theta^(-1/2)) for fixed alpha, answering the matching-lower-estimate question left by JoyboyBrian's merged #5103.

K[k,i] = alpha^-1 exp(-pi/alpha^2 (i-k/rho-c)^2)
rho=1+theta, c=1/(2rho), theta(N-1)<=1

||K^-1||_2 >= alpha m exp(-pi theta (2m-1)^2/(alpha^2 rho))
for every integer m>=1 with 2m<=N

0<theta<=1/16, N>=2 floor(1/(2 sqrt(theta)))
  => ||K^-1||_2 >= (alpha/4) exp(-pi/alpha^2) theta^(-1/2)

The new proof preserves a positive term of the checkerboard finite q-LDU inverse through the exact diagonal gauges, then applies an explicit alternating test vector to an m by m boundary rectangle. Combining this lower estimate with #5103's cited upper theorem proves matching order. The proposed sharp leading coefficient in Ryanson Jonathan's #4435 remains open. No new kappa or selected-frontier change.

This contribution was produced by Nick Petalas. Credit JoyboyBrian #5103 for the upper theorem and lower-bound question, Ryanson Jonathan #4408/#4435 for the scaling observation and formal prediction, Jose Manuel/jmbarrancoml #4193 for the finite inverse estimate, Vlado Vojdanovski and the retained #1673 Gaussian/q-LDU lineage, Cinna Davis's later caller, and the classical q-binomial identities. Predecessor code, numerical tables and operator replays are not the deliverable.

Evidence

Before: a rigorous theta^(-1/2) upper theorem with a matching lower estimate explicitly open. After: the quantified lower estimate above proves that order is necessary on the stated boundary sections.

The independent exact verifier passes 36 rational Gaussian matrices, 416 inverse-entry checks, 50 rectangle/test-vector checks, 520 rational exponent checks, 1,009 floor cases and nine boundary/sign/overstatement controls. Rational fixtures use the more general positive-exponent Gaussian family; the written proof specializes to the original pi/alpha^2 kernel. At the actual alpha=3, theta=1/400, N=400 fixture, a rational certificate gives a new inverse-norm lower bound exceeding 109/5. This is not its exact norm.

Using CPython 3.12.15 and only the standard library, both registered commands pass locally. Warning-strict normal and optimized direct runs take 0.79 and 0.80 seconds; read-only Git-free export runs take 0.80 and 0.81 seconds. The registered runner, six-file committed-byte/source-history preflight and official export integrity pass at head c16d2fc. Normal and optimized export reports agree, and the generated external receipt binds all six package files by SHA256.

From the repository root:

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

Any CPython 3.12 installation can replace the displayed interpreter path. Expected markers are PASS_GAUSSIAN_PHASE_CELL_LOWER, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Advance CLAIM. Scientific base a5df48c, priority revision 7. The uniform-transfer-audit card changes to this original conditioning side result. Main and governing guidance were refreshed unchanged before committing. Live PRs, handoffs and question-specific reservations were rechecked. The initial complete remote cache covered 305 open PRs and 4,166 Discussion records; sparse local-note coverage and unpublished work remain limitations.

Limitations and unrun obligations

The theorem fixes beta_0=-1/2 and needs a section containing a theta^(-1/2) boundary window. It does not give this lower order for tiny sections or sections separated from the cell edges. Its alpha-dependent constant is coarse; no sharp leading coefficient or matching growing-alpha law is proved.

The written argument carries the universal quantifiers. Finite exact checks corroborate it. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker, predecessor scientific replay or selected operator replay was run. Coefficient preparation, rounded inverse application, circuit prefixes, source access, restoration, storage/tape, native implementation, compiler and outer integer recovery remain unpriced. This is not a supplier admission or complete multiplication theorem. Every registered package command ran locally; none is deferred to CI.

Open next questions

• Recover Derive the broad forward inverse asymptotics: ||K^-1||_2 ~ e^(a/4)/(2 alpha sqrt(pi)) theta^(-1/2) from a turning-point harmonic oscillator; confirmed to 0.3% (no new κ) #4435's sharp leading coefficient with a uniform boundary remainder, including a growing-alpha regime. Start with more positive inverse terms and optimized boundary test vectors; a comparably sharp upper estimate remains necessary.
• Bind one complete rounded Gaussian caller and retain its coefficient, prefix, restoration, movement and storage costs before changing its numerical guards.

Related open work

No open PR studying this broad phase-cell inverse lower-order quantity was found in the live list and complete cached PR-body coverage. #5103, #4435, #4408 and #4193 are merged predecessors. Their upper estimates, formal prediction and constant-shift results are cited at their stated scopes; no additional computations were run to compare their reported numbers.

Completion refresh: this account has ten open PRs created since the advance CLAIM, including this PR. The nine others are #5145, #5146, #5148, #5149, #5151, #5152, #5153, #5154 and #5155. None studies this broad phase-cell inverse-norm quantity. Two adjacent precision results have different hypotheses and quantities:

• Prove sharp all-precision Neumann truncation at 226p² (no new κ) #5145 proves an optimal scalar Neumann truncation coefficient with its contraction and prefactor granted. This proves an inverse-norm lower order and does not compare that scalar tail or its constants.
• Prove sharp quadratic rounding guards for closed words (no new κ) #5153 proves an endpoint-conditioned coefficient-rounding guard for finite words. This concerns exact completed Gaussian conditioning and does not establish a rounding guard or compare its constants.

CI notes

Both hosted checks passed at c16d2fc11e2891bec18d6a9fa3f8ed4e743de6b2 in Registered research packages run 39943550152: plan in 3m35s and replay (research/gaussian-phase-cell-lower) / replay (research/gaussian-phase-cell-lower) in 1m21s. The run is completed with conclusion success, and its head SHA matches the contribution. Hosted replay executes the registered normal and optimized checks. All local checks listed above also passed; no registered check was deferred to CI. These automated checks do not establish independent human or formal proof-kernel acceptance of the written theorem.

Merge danger

Door: two-way. Blast radius: isolated. Six files are added in one new research package; shared code, frozen evidence and selected-frontier records are unchanged.