[Clear1nsight]

Change and mathematical scope

Prove a conditional integer-profile envelope at C=330007/100000000, rho=1e−9, c=8,b=9,u=1/5,v=1024. For every finite honest integer addition histogram A and actual target-loss subhistogram L, with P=P0+A, T=T0−L, the one strict mixed budget A_t+(1−rho)L_t<R0<43 implies alpha<1−rho, gamma,delta<.97, and alpha^1024+2gamma^1024+2delta^1024<1.

All 240 H25 baseline targets remain hypothetical. R0<43 is an explicit conditional premise, not a newly recomputed price or an accepted future ledger. No new targets, paid deletions or signed netting are permitted. Separate A/L quotas do not replace the mixed inequality, and any later 8H1 charges spend 8 from the same allowance. The rank estimate r<1849 supports a tail bound; it is not an ambient or physical geometry certificate. This positive scalar lemma supplies no funded κ or complete compiler/physical/ordinary/precision/stock/outer admission.

The PR also repairs the shared CI navigation document after the selected frontier promotion: its current section now points to #5413/#5407/#5414 and make frontier5413-verify, while the #4610 section links its historical checkpoint. This documentation-only successor changes no scientific package bytes or tests.

Task and overlap

Explicit donor assignment answers priority-revision 7 independent-moment with scope_status: changed for a positive robustness lemma. Canonical base is 1a3ced60b2f7a46bc70280a04f2eeb8ca027c700; selected κ remains .003027288235 under the same five assumptions. Peer precision approved this complementary publication and retains ownership of the extension. There is no overlapping planned robust-tail publication and no unreleased receiver or physical word in this package.

P0/T0 are retained from public PR5418, exact fe4c345778554ec5c02368b1ddf2f6918eeb0df5, with a raw Git commit/tree/blob origin proof. That public side commit is not claimed as a canonical ancestor; the needed model bytes are self-contained. Peer lemma c861887a57044497089b1861f2c6294a3e42409c48736c030776b7a8d7b7ec88 and root independent audit 2033e1d00fe41fbcdd15da84910296c3611a947c9175d8c4a0a12e3a44498d6c are LOCAL credit references by hash only. Root owns publication, live overlap and hosted CI.

Validation

The proof derives baseline bounds (T0)_t>1024320/65 and (P0)_(t+u)/8^u<301423/20 from finite binomial and integer-power comparisons. The mixed budget and integer multiplicity give A_t<43, L_t<44, r<1849<8·3^5, and added paid higher numerator below 129. Thus the remaining denominator exceeds 1021460/65 and

gamma <19760195/20429200<97/100,
delta <828360/1021460<97/100.

Their exact cross-product margins are 5612900 and 16245620. Bernoulli plus (100/97)^32>2 and the exact reserve threshold rho>1/1099511626752 close the full scalar envelope. The written proof quantifies all future A,L; finite certificate checks do not enumerate them or verify an actual future mixed budget.

The small verifier uses only integer/rational standard-library arithmetic, with no Powers, fractional-power evaluation, price point, native/receiver/compiler/primary engine or physical data. Nine meaningful domain/model controls reject fractional additions, negative netting, excessive losses, new targets, paid deletions, dropping the mixed hypothesis, ambient/earned-target overclaims and changed parameters. A=42H1,L=H4 is a valid integer-domain countercontrol: both loose separate quotas hold but its mixed consumption exceeds 43. Public-model byte and forged-origin mutations also reject. All normal/O predicates are explicit, source loading uses a private bytecode namespace and full pre/post custody is required.

Independent static review passed for the sealed 23-file snapshot and all 10 public origins. The sole peer completed exactly four integer-certificate executions: registered normal/O pair 1.410836020 seconds, readonly temporary Git-free normal 0.302753654 seconds, optimized 0.504951121 seconds, each under 30 seconds/128MiB with exit 0 and no resource stop. All four complete certificate JSONs are identical with zero comparison exclusions; their raw receipt hashes are also identical. Default sorted-JSON digest is 1c2bd9ecc56d6ff84808fdf7190e88885e121619ca920ad3676c132a8f47ec4d. All 23 working files and 23 tested readonly temporary files remain unchanged; both direct modes used empty PATH, files 0444 and no .git. Raw child receipts/stdout/stderr, exact guard logs and pre/post inventories are preserved.

Only after these tests, the identical 23 source files were committed as 302c11c02b15be8a62c8da29784b1bb11737cf1c, direct parent 1a3ced60b2f7a46bc70280a04f2eeb8ca027c700. Current metadata, committed-byte/declared-source-history and official export/check pass. Official export SHA256 60d1fab0d32a79c5b0a3323036d2367d08362cc6c4b97b9d3a854f8c81412f56, 634,880 bytes, contains 23 source files plus its transport receipt. Every exported source byte matches the reviewed/requested/executed working and readonly temporary source. The final official 24-member export was byte-checked through this explicit bridge and was not separately rerun. During local assembly there was no pricing/Powers, physical, primary or additional integer-verifier execution; the subsequent hosted replay is reported separately below.

Hosted CI at original head 302c11c02b15be8a62c8da29784b1bb11737cf1c failed in shared onboarding test CurrentCiNavigationTests.test_current_ci_section_matches_selected_record before package selection or replay. The CI guide still named make frontier4610-verify while the selected record required make frontier5413-verify. The unchanged six-test navigation module reproduced that failure before the documentation repair and passed all six afterward. The current publication head is baa8fbb8345c2fab0bc213479539c338d4c80443, whose sole delta from the tested scientific commit is docs/ci-verification.md. All 23 mathematical package blobs remain exactly identical. No tests or workflow checks were disabled, and no extra local scientific replay was run. Hosted run 40062451036 subsequently passed at this exact successor head: shared onboarding/selection and registered package replay, including committed-inputs-unchanged. The earlier failed run remains preserved as evidence. These hosted executions are separate from the four local certificate runs above, and do not execute the later written analytical addendum or establish physical or kappa admission.

python3 -B scripts/research_package.py --package research/kappa0033-robust-tail-envelope
python3 -B scripts/check_research_commit.py --package research/kappa0033-robust-tail-envelope --check-source-history

Direct Git-free command: /usr/bin/python3 -I -B research/kappa0033-robust-tail-envelope/verify.py; add -O or use --integrity-only to authenticate without running the integer verifier. No third-party dependencies. Source/history/export checks establish declared provenance and transport, not complete local canonical history, human/kernel proof or hosted-CI equivalence.

Credit

Peer precision owns the robust-tail extension; root supplied the elementary baseline derivation and independent written audit. Public PR5418 supplies the explicit profile model and motivating conditional bound. Preserve construction/resource owners, Dillon, Ali Romman, Cao Yifan/native/Clifford, Cinna Davis, zxv, Jacob Alan Sussman, Dugongue, Chris R/Clear1nsight and all native/helper/compiler/precision/analytic/recovery notices. New exact finite certificate checking, source custody and publication packaging by Chris R / Clear1nsight, with substantial OpenAI Codex assistance; Apache-2.0. No new construction priority or independent human acceptance is claimed.

Open next questions

A separate self-contained written addendum derives total gross added rank mass at most 43 and the aggregate-target-moment-loss extension under its stated count and budget hypotheses. It was independently reviewed as written mathematics, is outside this sealed verifier, and claims no new execution, actual caller funding or private construction.

A future actual receiver must bind its complete honest gross A,L account and prove the one strict mixed inequality, with all extra H1 calls and target losses included. Canonical independent outputs, arbitrary owned/restoration columns, ambient/adapters, ordinary movement/cost, precision/error/semantic constants, stock/tape, chronological splitting/completion and full compiler/outer application remain separate obligations. The lemma removes repeated scalar tail checks only within its exact hypotheses. A changed target profile that increases some rank multiplicities, including a rank-downgrade replacement, requires an explicit transfer argument outside this stated subhistogram theorem; no such transfer is claimed here. The goal of achieved conditional κ≥.0033 remains open; TASK.json records relevance, first steps and blockers.

Scientific handoff

HANDOFF

Research discussion / PR / artifact: Local research/kappa0033-robust-tail-envelope, a peer-approved complementary scalar envelope using public PR5418 profile data. Operational packet outputs/kappa0033-robust-tail-envelope-20261024. Root owns the public PR, live overlap, exact-head readback and CI. No unreleased receiver or physical word is included.

Exact commit: 302c11c02b15be8a62c8da29784b1bb11737cf1c, branch research/kappa0033-robust-tail-envelope, direct canonical parent 1a3ced60b2f7a46bc70280a04f2eeb8ca027c700. Public PR5418 model source fe4c345778554ec5c02368b1ddf2f6918eeb0df5 has separate raw Git origin and is not claimed as canonical ancestry. 23 package files; export SHA256 60d1fab0d32a79c5b0a3323036d2367d08362cc6c4b97b9d3a854f8c81412f56, 634,880 bytes. Metadata, committed-byte/declared-source-history and official export/check pass. REQUEST-EXECUTION-COMMIT-EXPORT-BRIDGE.json closes every reviewed/requested/executed/working/committed/exported source byte. It establishes declared base/direct-parent and transport custody, not complete local canonical history.

Current PR publication head: baa8fbb8345c2fab0bc213479539c338d4c80443. It adds only the CI navigation repair above to scientific commit 302c11c02b15be8a62c8da29784b1bb11737cf1c; the 23 package blobs, four local certificate receipts and original scientific export remain unchanged. The shared onboarding failure preceded package replay, the six unchanged targeted tooling tests now pass, and successor hosted CI run 40062451036 passed both plan and registered replay, including unchanged inputs. This hosted evidence is separate from the four local certificate executions and does not validate any physical supplier or the later written addendum.

What is established: Under fixed C=.00330007, rho=1e−9, c=8,b=9,u=1/5,v=1024, one honest mixed budget for P=P0+A,T=T0−L implies alpha<1−rho and gamma,delta<.97, hence the scalar three-moment envelope below 1. The written proof quantifies every finite nonnegative integer A and subhistogram L, excluding new targets, paid deletions and signed netting. Exact finite binomial/rational certificates support baseline bounds 1024320/65 and 301423/20, remaining denominator 1021460/65, added numerator 129 and tail cross-product margins 5612900/16245620. Nine meaningful domain/model controls plus the 42H1+H4 trap reject replacing the mixed inequality with separate allowances. Independent static review passes. The sole peer ran all 3 requests once: registered normal/O pair 1.410836020 seconds, readonly temporary empty-PATH normal 0.302753654 seconds and optimized 0.504951121 seconds, under 30 seconds/128MiB, exit 0/no resource stop. All 4 complete certificate JSONs and raw receipt bytes are identical with zero exclusions; default sorted-JSON digest 1c2bd9ecc56d6ff84808fdf7190e88885e121619ca920ad3676c132a8f47ec4d. Both 23-file working/readonly inventories and modes remain unchanged. Raw actual receipts, child stdout/stderr and guard logs are preserved.

What remains uncertain: Actual future complete P/T ledger and strict mixed budget; independently earned canonical outputs; ambient geometry/adapters; owned entry/returns/restoration; ordinary cost/movement, precision/norm/error, stock/tapes, full compiler and outer/recovery application. All 240 H25 targets remain hypothetical, R0<43 conditional, and r<1849 is not an ambient certificate. Selected κ=.003027288235 remains unchanged. The official export contains 24 members including its transport receipt, was checked and byte-bridged to the 23 executed temporary source files, and was not separately rerun. No Powers, pricing, physical/native/primary engine, funded κ, human approval or formal-kernel proof is supplied. No later rank43 refinement is included or tested in this sealed package.

Smallest useful next task: Root completes the independent changed-evidence review, then publishes the complementary support PR and checks exact head/CI. A future actual receiver must bind honest A_total and L to the one strict mixed budget, counting every later 8H1 charge against the same allowance. Separate physical/ordinary/precision/stock/compiler/outer joins remain required. No further verifier or source edit is needed for this packet.

Reproduction command and expected runtime: python3 -B scripts/research_package.py --package research/kappa0033-robust-tail-envelope runs only the exact integer/rational verifier in normal/O; actual guarded pair 1.411 seconds. Direct /usr/bin/python3 -I -B research/kappa0033-robust-tail-envelope/verify.py, add -O; readonly temporary empty-PATH times 0.303/0.505 seconds with files 0444 and no .git. --integrity-only authenticates without the integer verifier. python3 -B scripts/check_research_commit.py --package research/kappa0033-robust-tail-envelope --check-source-history checks declared provenance/transport. RUNREQUESTS-FROZEN.json, RUN-RESULTS.json, runs/, fresh/ and pre/post inventories bind actual commands, limits, full outputs and custody. Commit/official export followed the four tests with exact byte identity and no extra execution.

Predecessor credit: Peer precision owns the robust-tail extension; root supplied baseline derivation and independent written audit, retained as LOCAL hashes only. Public PR5418 supplies the baseline model and motivating conditional bound. Preserve construction/resource owners, Dillon, Ali Romman, Cao Yifan/native/Clifford, Cinna Davis, zxv, Jacob Alan Sussman, Dugongue, Chris R/Clear1nsight and all native/helper/compiler/precision/analytic/recovery notices. New exact finite certificates, source custody and packaging by Chris R/Clear1nsight with substantial OpenAI Codex assistance; Apache-2.0. The sole peer executed the four new integer certificates. No construction priority or independent human approval is claimed.

[Clear1nsight]

Analytic addendum: total paid-rank mass and aggregate target-moment loss

This written addendum complements the conditional scalar envelope in PR #5420. It supplies two analytic deductions: a total gross added rank-mass cap, and a target-moment formulation that does not require entrywise target deletion. These deductions are separate from the sealed #5420 package and were not included in or checked by its verifier. No new price evaluation, integer-check execution, physical replay or funded κ is claimed.

Setting

Fix

C=330007/100000000, t=1-C, rho=1/1000000000,
c=8, b=9, u=1/5, v=1024.

For a finite histogram X write X_s=sum_r X[r] r^s. Ranks are positive integers and job multiplicities are nonnegative integers. Let P0,T0 be baseline paid and target profiles with the elementary bounds

(T0)_t >1024320/65,
(P0)_(t+u)/8^u <301423/20.

These are the baseline bounds used in #5420. Here they are explicit hypotheses, rather than a newly evaluated numerical profile.

Let the final paid profile be P=P0+A, where A records all gross added paid jobs. Paid deletions, negative weights, fractional job multiplicities and signed paid netting are outside this representation. Let T be a finite final target histogram, with actual productive count

B=sum_r T[r] <=1416.

Define the aggregate moment loss at this fixed exponent by

Delta=(T0)_t-T_t,
R0=(1-rho)(T0)_t-(P0)_t.

Assume Delta>=0 and the one strict mixed budget

A_t+(1-rho)Delta < R0 <43.

Delta need not be the moment of a nonnegative loss histogram. The final T need not be entrywise bounded by T0. Its actual count, aggregate moment comparison and complete paid binding must nevertheless be established; labels or a counterfactual allowance do not supply those facts.

1. Total gross added rank mass is at most 43

Nonnegativity in the mixed budget gives A_t<43. The exact comparison

330007*256=84481792<100000000

gives C<1/256. Four positive binomial terms give

(44/43)^256
 >1+256/43+32640/43^2+2763520/43^3
 =4719891/79507 >44,

because 44*79507=3498308 and the difference is 1221583>0. Thus

44^C <44^(1/256)<44/43,
44^t >43.

Every present added job has multiplicity at least one. If its rank were at least 44, its contribution would be at least 44^t>43, contradicting A_t<43. Hence every present rank is at most 43.

For nonempty A, define its total gross rank mass by M_A=sum_r r A[r]. Then

M_A=sum_r A[r] r^t r^C
   <(44/43) A_t <44.

M_A is an integer, so M_A<=43. Empty A gives M_A=0 directly. Equivalently, finite power subadditivity gives M_A^t<=A_t; M_A>=44 would contradict 44^t>43.

The cap applies to all additional paid jobs together. Already incurred additions spend from the same total; the deduction creates no fresh allowance. It is necessary, not sufficient for the strict mixed budget, and it proves no physical ambient dimension or geometry.

2. Aggregate loss suffices for the same scalar tails

The mixed budget gives

A_t<43,
Delta<43/(1-rho)<44.

The second comparison follows from rho<1/44. Consequently

T_t=(T0)_t-Delta >1024320/65-44=1021460/65>0.

For an added rank r, the original coarse implication sqrt(r)<=r^t<43 gives r<1849<8*3^5. Hence (r/8)^(1/5)<3 and the added higher-paid numerator is below 129. Empty A also has numerator zero below 129. Thus

P_(t+u)/8^u <301423/20+129.

Define

alpha=P_t/T_t,
gamma=P_(t+u)/(8^u T_t),
delta=9B/(8^C T_t).

The denominator is positive, B<=1416, and 8^C>1. Therefore

gamma <19760195/20429200 <97/100,
delta <828360/1021460 <97/100.

The exact cross-product margins are respectively 5612900 and 16245620. The mixed budget itself gives

(1-rho)T_t-P_t=R0-A_t-(1-rho)Delta>0,
alpha<1-rho.

Bernoulli's inequality gives (1-rho)^1024<1/(1+1024rho). Also

(100/97)^32 >1+96/97+496*9/97^2 >2,

so (97/100)^1024<2^(-32). Since rho>1/1099511626752,

alpha^1024+2gamma^1024+2delta^1024
 <1/(1+1024rho)+4/2^32 <1.

This is the same conditional scalar envelope. The proof used the nonnegative aggregate Delta and the actual count bound, not entrywise target domination or integrality of Delta.

3. Sufficient matching and actual-census alternatives

A nonduplicating matching of baseline target slots to final targets, with each retained rank no larger than its matched baseline rank, is one sufficient way to establish the count and moment conditions. Deletion may be represented by rank zero, with 0^t=0. Such a matching can increase a lower-rank histogram coefficient while reducing a higher-rank one, so it need not produce an entrywise subhistogram.

This matching is not necessary. Changed roles or some increased ranks can instead be handled by a directly proved complete final census satisfying B<=1416 and Delta>=0. A role change alone establishes neither eligibility nor an obstruction. Conversely, a symbolic histogram comparison alone does not prove actual canonical independent outputs, a legal charged correspondence or the complete paid ledger.

Negative aggregate loss or a larger final productive count falls outside this version of the result. The total paid-rank cap, scalar inequalities and coarse rank estimates certify no actual physical endpoint, ownership/restoration, geometry, ordinary cost, precision/stock/tape, compiler or outer/recovery application. The achieved κ objective remains open.

Credit and verification status

The peer precision contribution supplied the robust integer-profile envelope. Root proposed the sharper paid-rank and aggregate-loss deductions; independent written checks reviewed their arguments and scope. Retain the public model, construction/resource owners and native/helper/compiler/precision/analytic lineage credited by #5420. This generic addendum was prepared with substantial OpenAI Codex assistance.

No private construction or application is included. These written deductions change no sealed source, selected result or execution receipt and claim no independent human approval or formal-kernel acceptance.

Scientific handoff

HANDOFF

Research discussion / PR / artifact: This self-contained analytical addendum to PR #5420, published as a separate written result. It is outside the sealed executable's verification scope.

Exact commit: The sealed mathematical source is 302c11c02b15be8a62c8da29784b1bb11737cf1c. Current publication head baa8fbb8345c2fab0bc213479539c338d4c80443 adds only CI navigation documentation and preserves all 23 package blobs. This comment creates no new source commit or execution receipt.

What is established: The displayed proof gives total gross added integer rank mass at most 43. Under actual final target count B<=1416, nonnegative aggregate moment loss and the one strict gross-integer mixed budget, the same scalar alpha reserve and gamma/delta bounds follow. Separate written reviews found no substantive error. Entrywise target deletion is unnecessary; a rank-downgrade matching is only a sufficient application rule.

What remains uncertain: Actual physical construction, independently earned canonical targets, complete paid/target ledger and strict budget, all ownership/restoration/geometry/ordinary/precision/stock/tape/compiler/outer obligations. No new price, solver or integer-verifier execution validates this addendum; no funded kappa, independent human approval or formal-kernel proof is claimed. Hosted CI remains a separate gate for the sealed package and documentation successor.

Smallest useful next task: Supply one actual caller's independent target census and honest gross paid ledger, verify the count/moment/budget hypotheses, and then discharge the physical and source-application gates. A role change alone supplies neither eligibility nor an obstruction.

Reproduction command and expected runtime: Review the self-contained finite integer comparisons and real-power inequalities above. No scientific execution is prescribed. The original #5420 verifier checks only its declared sealed theorem and supporting constants; it does not check these added deductions.

Predecessor credit: Peer precision supplied the original robust integer-profile envelope. Root proposed the rank-mass and aggregate-loss deductions; separate written AI-assisted reviews checked them. Preserve the public model and construction/native/helper/compiler/precision/analytic lineage credited by #5420. Prepared with substantial OpenAI Codex assistance.