[petalas]

Summary

Scoped proof with exact finite premises and a sharp equality certificate. A common real ambient deformation of any two-disjoint-frame puncture of E8 satisfies the new quantitative identity

A0 = A - tr(A) I/8
E_P(A) = 37 ||A0||_F^2
         + (17/2) sum of squared deleted-frame diagonals
         + sum of squared orthogonal cross-frame defects.

Here A is symmetric and E_P sums squared absolute inner products over unordered orthogonal unit-root pairs of the 104 remaining lines. Therefore E_P(A) >= 37 ||A0||_F^2. An explicit rational symmetric involution has norm squared 8 and energy 296. Rational invertible common maps I+sA attain equality arbitrarily near identity, proving 37 sharp even for positive definite metrics from rational maps.

For M=T^T T this bounds distance from a conformal metric by sqrt(E_P(M)/37). Exact qualitative rigidity is prior work; the contribution is the quantitative sum-of-squares identity and sharpness family. No new kappa, supplier admission or selected-frontier change is claimed. Six files are added under research/p104-metric-energy/.

Produced by Nick Petalas. Credit Brett Halligan / #1212, Aaron Wang / #1292, Josie Elliston / #1389 and #1399, and their retained E8 decoder lineage for the prior exact-realization questions and qualitative results. Standard E8 root coordinates and elementary frame facts retain their mathematical credit. No predecessor executable or proof is replayed as the deliverable.

Evidence

• Before: qualitative exact-rigidity results supply context, without being re-audited here. After: the new identity gives an explicit defect bound and an attaining family.
• Exact integer checks enumerate all 120 root lines, 3,780 full orthogonal pairs and all 120 tight stars. Every quadratic identity is checked on all 1,296 coefficients. The equality example enumerates 2,860 orthogonal pairs and has energy 296. Seven controls distinguish missing data, invalid frame boundaries, a changed witness and a stronger proposed constant.
• Warning-strict normal and optimized direct runs and both registered commands pass on CPython 3.12.15, macOS 27.0.1 arm64. Initial timed direct commands each took 0.13 seconds, below the registered 60-second limits.
• Committed bytes and declared source ancestry pass at 5badad3af2ff7f6c325f20813a6176cc7e3111c0. The official export and input-coverage check pass. Normal and optimized read-only Git-free replays with empty PATH agree; all seven exported files retain their hashes.

Exact reproduction from the worktree root, using the local interpreter:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-metric-energy/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p104-metric-energy/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-metric-energy
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-metric-energy --check-source-history

Any CPython 3.12 interpreter may replace the absolute path. The direct checker prints PASS_P104_METRIC_ENERGY and a generated receipt bound to all six source files.

Limitations and unrun obligations

The written frame-removal argument supplies the quantifiers over every pair of disjoint frames and every real symmetric matrix; the checker supplies exact finite E8 premises and the explicit equality example. This is a common-metric result using absolute inner products before separate normalization. Independent port motions, arbitrary masked fitting factors, normalized correlation optimization and complex Hermitian deformations are outside its scope. The sharp example is not identified with the selected P104 frame-pair orbit.

No native implementation, dirty restoration, complete signed invoice, selected supplier replay, all-size compiler, precision/tape proof, complete multiplier, Lean or Docker run is supplied. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. The card's change to this geometry boundary question is explicit in TASK.json.

Open next questions

• Determine the sharp constant for each individual disjoint-frame orbit, especially the selected P104 orbit. First compute the kernel of the residual squares for a fixed representative.
• Test independent port motions outside the common-metric class. First give one arbitrary-input local identity with restored outputs, an actual consumer and its complete paid/productive bill.

Related open work

• Construct an exact dyadic rank-seven P104 projector (no new kappa) #6247 constructs a dyadic complex rank-seven P104 projector; this package studies real common-metric orthogonality energy, a different error quantity.

• Add P104 owned-lineage factorization and scoped review of PR6170's owned-layer floor #6267 studies P104 owned-lineage and decoder cost floors; no owned-layer cost is counted here.

• Prove correlated dirt is necessary at P104 wire meetings (no κ) #6261 studies correlated dirt at a P104 wire meeting; this package has no dirty-wire or temporal model.

• Remove one scalar ADD from the selected P104 word (no κ) #6251 changes one scalar ADD in the selected P104 word; no scalar word is changed here.

• Prove the P104 reader-497 two-cache cost floor (no new kappa) #6248 studies reader-497 cache cost; this package has no cache or reader tariff.

• Conditional κ 0.00309871 from a prepared P104 split helper #5606 proposes a P104 split-helper supplier; this geometry theorem does not admit or reprice it.

• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 proposes P104 successor handoffs; their physical words and conditional bounds are outside this result.

• Checkpoint symmetric two-star P112 supports and inconclusive COPY63 search #5088 searches P112 two-star decoder supports; no support dictionary or COPY63 search is extended here.

• Checkpoint eight masked-decoder basis searches (exploratory; no new K) #5064 searches masked E8 decoder completions and factor bases; general fitting factors remain outside the common-metric class.

• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies binary fitting-pairing and square root-code decoder floors; this package has no binary term-cost claim.

• Rank-8 rigidity of the E8 pattern with arbitrary extra zeros after deleting any at most 8 lines (hypothesis (1) of Lemma RIGD; no kappa claim) #3040 studies qualitative rank-eight fitting rigidity after at most eight deletions; this package gives quantitative energy after two frame deletions.

• Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 proposes a dirty-restored P104 gather/channel; no such program or native boundary is assumed here.

• Add P104 flag 0 coefficient transport audit #3915 studies P104 coefficient transport; this result has no chronological transport claim.

• Research P104 singular-mixing lifetime floor #3907 studies P104 singular-mixing lifetimes; this result has no lifetime model.

• research: price full P104 end-cut correction #3894 prices a P104 end-cut correction; this package supplies no correction invoice.

• Isolate explicit historical receipt adapters from P104 research #4103 isolates historical P104 receipt adapters; its shared-code changes are separate from this new self-contained checker.

• Exclude saturated Gram systems at every even window h >= 6 (no κ) #6268 excludes saturated even-window Gram line systems; it studies label-density equality rather than common-metric defect energy on a fixed P104 puncture.

• Prove the exact radius-three full recovery fiber threshold (no κ) #6269 determines a bounded-input recovery threshold for a fixed rounded native map; no rounded recovery map is analyzed here.

• Reduce complete mapped-contact replay time by 72.54% (no new κ) #6274 accelerates a mapped-contact rank replay; it does not change or compute this real metric-energy identity.

• Prove the sharp rank-only P104 support floor 96 (no κ) #6275 determines the rank-only P104 owned-delivery support minimum and its extremizers; its support-count objective differs from this orthogonality energy.

• Add division-free dual-pair exchange: 134 to 64 scalar units (no κ) #6280 constructs a division-free dual-pair exchange with a spare role; this package supplies no such scalar compiler or physical helper interface.

The completion refresh inspected this account's open PRs created since the advance CLAIM. No other inspected contribution computes this common-metric defect-energy constant. No same-quantity numerical disagreement was identified; no extra computation was run to validate other reported numbers.

CI notes

Both hosted checks passed at head 5badad3af2ff7f6c325f20813a6176cc7e3111c0: plan and replay (research/p104-metric-energy). Run 41067782349 completed successfully at the matching head. The completed checks were read once; run metadata separately confirms the head, completed status and successful conclusion. All local checks described above also passed.

Hosted replay certifies the registered finite checks and package controls. The quantified sum-of-squares argument remains a written scoped proof; independent human and formal proof-kernel review are unclaimed. The delivered scope matches the advance CLAIM.

Merge danger

Two-way door. Blast radius: isolated. Only the new six-file research package changes; shared code, frozen evidence and selected-frontier records are untouched.