[petalas]

Result and claim type

Finite certificate; no new κ. Ali Romman's #5838 proves a 20-bit completed-operation rounded-error guard for the PR5830 P104 gauge word. Its second Open next question asks for the bound to be recomputed for a later gauge tradeoff word. Ali Romman's #5833 emitted two such complete words, with sharp ideal amplification 96 (original scalar budget) and 2 (minimum), and left their rounded pricing open. This package applies #5838's contract and causal bound, unchanged, to both words.

Word	Ideal peak	Causal guard	Final public error	Intermediate error	Field norm + error
PR5830 (#5838's result)	65,537	20	< 0.97754 B	< 26.903 B	< 65,563.91 B
#5833 budget word	96	20	< 0.97746 B	< 0.97746 B	< 96.978 B
#5833 minimum word	2	20	< 0.97746 B	< 0.97746 B	< 2.978 B

• The final guard stays at 20 bits. At guard 19 the causal bound on both new words is above 1.955 B.
• The intermediate error envelope falls from 26.9 B to below 0.978 B. On both new words the largest error bound anywhere is the final bound on public field 205.
• The field-magnitude envelope falls from below 2^17 B to below 97 B and 3 B. Read as fixed-point headroom (an inference, not a physical proof), that is 7 and 2 integer bits above B instead of 17, at the same 20-bit guard. The budget word costs 24,332 more scalar primitives than PR5830's 200,546 and still fits the original 224,880 allowance.

Every physical MOVE, ADD and SCALE is a defect site. The #5833 words already contain all entry, exit and lowered SCALEs and have no identity SCALE, so no site is virtual. As an adapter control, PR5830's emitted word under this convention reproduces #5838's recorded 20-bit causal case exactly: every final coefficient, the peak value and field, and all 26,271 sites. Only the peak's instruction index differs, because #5838 also numbers exact virtual identities. This checks the site convention; it is not a re-check of #5838.

CLAIM. The delivered scope matches it.

This contribution was produced by Nick Petalas. Credit Ali Romman #5838 for the contract, causal bound, transfer method and PR5830 result, and #5833 for the two words and their sharp peaks, with the #5819/#5826/#5831 transfer and #5830/#5824/#5822/#5812 gauge lineage they retain. Independent human and formal proof-kernel review are unclaimed.

Evidence and reproduction

Commit 2acae34be8b894b34fc3953088c7489c51059f2b, base canonical main 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. Six live inputs (#5838's causal.py, transfer.py, RESULT.json and PR5830 word; #5833's two words) are pinned by Git blob and SHA256.

From the repository root:

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

Use an equivalent CPython 3.12 elsewhere. Sparse checkouts need research/p104-gauge-error-covariance and research/p104-priced-amplification.

Local CPython 3.12 / macOS arm64:

• Warning-strict normal and optimized runs print PASS_P104_GAUGE_WORD_ROUNDED_GUARDS, about 17 s each. The verifier requires the committed RESULT.json to equal its recomputation byte for byte.
• The registered command passes PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed in 20.6 s wall (CI limit 600 s).
• check_research_commit.py --check-source-history passes PASS_RESEARCH_COMMIT_BYTES.
• Four controls are rejected: a truncated word and a doubled SCALE coefficient fail the endpoint check, guard 19 is not accepted, and omitting 1,000 sites changes the bound.

Limitations and unrun obligations

• This is an upper bound under Prove the sharp 20-bit completed-error guard and gauge-invariant obstruction #5838's worst-case contract. Sharpness of 20 bits is not claimed for the new words. Prove the sharp 20-bit completed-error guard and gauge-invariant obstruction #5838's MOVE/ADD-only 19-bit trace survives every field-constant signed scalar gauge of its base word, but I did not check that the Construct P104 at the sharp amplification floor and optimal original-budget peak #5833 words lie in that class.
• Hardware rounding with stronger correlations may need fewer bits.
• No rounded native child, COPY/reset machine, signed-record range, tape or descriptor cost is built or charged. The five framework assumptions remain. No κ, selected-frontier or admission change follows. No Lean build or full multiplication replay was run.

Related open work

• Independent replay, invoice, native and funding audit of #5827 and #5907 (no new κ) #5960 (Matarener) audits Conditional κ = 0.00312476680 (+5.8046936e-7 over #5811): unread path #5827 and Conditional κ .003125282 (+5.22e-7 over #5827): remove helper 468's frame-593 rung with rerouted reads #5907, which are different P104 words. It does not compute a rounded guard for the Save 438012 P104 scalar primitives with a complete optimal gauge construction #5830/Construct P104 at the sharp amplification floor and optimal original-budget peak #5833 words, and no number is shared.
• No other open PR found studies rounded-error guards on the Save 438012 P104 scalar primitives with a complete optimal gauge construction #5830 or Construct P104 at the sharp amplification floor and optimal original-budget peak #5833 words.

Completion refresh inspected this account's open PRs created since the CLAIM: #5963, #5965, #5966, #5967, #5968, #5969, #5970 and #5971. Only #5969 concerns a P104 word; it counts scalar units after sign normalization and computes no rounded-error guard or field envelope, so no number is shared. No computation was run to check another contribution.

Open next questions

1. Is 20 sharp on the Construct P104 at the sharp amplification floor and optimal original-budget peak #5833 words? Transport Prove the sharp 20-bit completed-error guard and gauge-invariant obstruction #5838's FAMILY-ADVERSARY.json, or show the words are field-constant signed scalar gauges of its base word. The first step is to map each Construct P104 at the sharp amplification floor and optimal original-budget peak #5833 instruction to its base event and gauge ratio.
2. Does the narrower envelope buy a cheaper physical word? Price 27-bit against 37-bit fixed-point records in an actual tape/descriptor tariff, including the budget word's extra 24,332 scalar primitives.
3. Can any gauge lower the final guard? Both new words keep the final bound near 0.9775 B, so amplification does not bind it. The first step is the per-site share of field 205's final bound.

CI notes

At commit 2acae34be8b894b34fc3953088c7489c51059f2b, run 40539806367 passed both hosted checks: plan and replay (research/p104-gauge-word-rounded-guards).