Independent verification: complete U8 terminal-cone census (no new kappa)

research/u8-terminal-cone-census/COST.md

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+# Complete optimistic recurrence and its limits
+
+Let `d_k` be the literal end-B dimension, `u0_k` the actual old terminal
+dimension and `s_k` the entrance dimension. A proposed helper endpoint has
+dimension `u_k`, contains its necessary cone, and has width `u_k-s_k`.
+Retain all source/target endpoints, the eight ordered centers, A/B frame
+assignments, nine calls and 896 live families per replica in ambient rank73.
+
+The exact old local histogram is
+`{1:2548,2:338,3:446,4:162,5:72,8:8}`. Its decomposition is:
+
+| Bill | Rank histogram | Rank mass |
+| --- | --- | ---: |
+| Fixed prefix climbs and center descents | `{1:2048,2:288,3:368,4:160,5:16,8:8}` | 4512 |
+| Actual old post-B suffix, all728 registers | `{1:500,2:50,3:78,4:2,5:56}` | 1122 |
+| Favorable single jump to each old endpoint | `{1:328,2:80,3:86,4:10,6:56}` | 1122 |
+
+The first two sum to the full old profile of mass5634. The favorable
+single-jump relaxation preserves mass, but can reduce its moment. It need
+not admit any common-frame cleanup word. This note makes no numerical
+exclusion from that relaxation.
+
+In particular, removing an assumed `H(9-d_k)` and adding `H(u_k-d_k)` is
+unsafe as a general rejection test. Old endpoints need not be FULL, and
+post-B climbs may be split across early cleanup. If other registers' suffix
+climbs may change, retaining their entire old bills is also unsafe. Use the
+actual old suffix or replace **all** 728 suffix bills by favorable one-jump
+costs, including sources and targets.
+
+The fixed source/target/center mass is1856. The old helper widths are
+`{4:80,5:32,7:54,8:122,9:216}`, of total3778. The capacity mass for idle
+stock per replica is48384. For proposed widths set
+
+```
+D(u) = 48384 - 9 * sum_helpers(u_k-s_k)
+     = 14382 + 9 * sum_helpers(u0_k-u_k).
+```
+
+Require `D(u)>=0` for feasible whole-helper placement. Every idle child has
+rank at most48. For `0<z<1`, define `phi_z(r)=r*(73/r)^z`, `phi_z(0)=0`.
+Concavity gives `phi_z(a)+phi_z(b)>=phi_z(a+b)`. With P_pre the fixed prefix
+histogram, a complete **optimistic moment lower bound** is
+
+```
+L_z(u) = [9 * (sum_r P_pre(r)*phi_z(r)
+                 + sum_all728 phi_z(u_k-d_k))
+           + D(u)*phi_z(48)/48] / (73*896).
+```
+
+It includes every required suffix dimension and charges returned helper
+mass as idle work. Ignoring bin integrality, shared-gate compatibility,
+chronology and new finite scalar work enlarges the candidate class. A
+certified minimum `L_z>=1` would exclude an improved complex endpoint beyond
+z for this precise family; `L_z<1` is inconclusive. No such outward-rounded
+numerical exclusion is asserted here. The endpoint census alone forces
+every reductions-only helper proposal back to its old endpoint.
+
+For an actual program and T replicas, the exact recurrence must instead use
+the freshly recounted local paid histogram L_new and **literal** idle
+histogram I from a valid whole-helper owner map:
+
+```
+H(r) = 9*T*L_new(r) + I(r)
+M = 896*T, W = 73
+sum_r r*I(r) = T*D(u)
+M*W - sum_r r*H(r) = 320*T
+```
+
+The last identity follows from local mass `1856+sum_helpers(u_k-s_k)`;
+width reduction does not erase the inherited rank deficit. The associated
+weighted recurrence uses these children plus a separately charged finite
+term. That term must recount compensation, scalar/Horner coefficient
+access, frames, copies, wrappers, source restoration, routing, bridges,
+shifts, moves, private stock, correction streams, precision, tape and rows.
+Keep the bit supplier and its surcharge, analytic guards and seven-step
+outer recipe. Zero extra finite work is allowed only in the relaxation,
+never in a constructed invoice.
+
+A positive nomination therefore still needs a literal cleanup, both exact
+arbitrary-dirt orientations, native phases and wrappers, changed residual
+adapters, complete whole-helper hosting, the full invoice and rational
+moment/outer certificates. This contribution changes none of those inputs
+and claims no larger kappa.
+
+Prepared for ToikY with substantial OpenAI Codex assistance. Apache-2.0.

research/u8-terminal-cone-census/MANIFEST.json

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+{
+  "CERTIFICATE.json.gz": "980ce3d07d20a6f1bb1c9d097f91aa38b08d19681160f025653e2019f634a95d",
+  "COST.md": "4583754502b150ee5593b1fc9295c7c5785e26a179bd5f61013a018487b3dd7c",
+  "PROOF.md": "a4419c30370ab74cb7c20125a7d166710e368f47a46e4c4fbc710df85bd6e879",
+  "README.md": "461f01830ade53f779fb4a5fbc871fa90efabda378964bfe2d827bebf78c91c3",
+  "SOURCES.json": "df52be5641c2310b9bff76c742f8fb420605ab389658227c29e44a16676dfe95",
+  "audit/COMMAND-1.json": "be259843550ef3110093af0e938bc9dbd790bf17684448cde4f452afbb0d37e2",
+  "audit/COMMAND-2.json": "c21df59dcdfe4dd380cb065a650fc353749627dd369514c9094929b15e1022db",
+  "audit/INDEPENDENT-AUDIT.json": "18431a3ca2f0c909f08a851770ce0612d33d9ef625df42ef6ab23fe159ab1acf",
+  "audit/INDEPENDENT-PROFILE-CHECK.json": "e7698c26e6e7c4a75f228110d42cfed5e4a673ef43c9a8a883e004a94aacdee8",
+  "audit/LATEST-MAIN-DEPENDENCY-CHECK.json": "9edf16266319115d76165540072342ee8895b3f11294802892637dd7ea0cfb1e",
+  "audit/NORMAL-PACKAGE-COMMAND.json": "6a0b6faee1128b9d18d754b6623b3de2a1b307e7d4ec4eba86c3875f52c9f853",
+  "audit/NORMAL-PACKAGE-OUTPUT.txt": "96253cd2a8d6d422269de6dc700861230a2c28dc630385c55fa385fdd4f5a83f",
+  "audit/PACKAGE-INDEPENDENT-REVIEW.json": "72410a6c7aa154c7debd186b71b201afcda6ed6cb2759b68ed3417677960ebdc",
+  "audit/PACKAGE-NEGATIVE-CONTROLS.json": "8b22a06da96e0b6ad91316387fb1d3f644c4e41f5e7a8557b95ed67964a9cc4c",
+  "audit/RELOCATED-OPTIMIZED-COMMAND.json": "bff7f4907ea6f49d31f745a115959558bb0433eb806d7e5d6ea129c111ca8111",
+  "audit/RELOCATED-OPTIMIZED-OUTPUT.txt": "092cebe53ac9ab45aa5595a87f5a6003a07a162f48b8846ccfa98af38a40b4e7",
+  "audit/REPORT-FINAL-OVERLAP.md": "e8432806389e1bda8cda13132b6ebe2d35de30cc1e91ad9fa63d451bea028eff",
+  "audit/SEMANTIC-MUTATION-CONTROLS.json": "0f9bda4894c26975a39c4a64df150d076e86a074da80ede528dc72989253dd41",
+  "audit/SOURCE-PIN-CONTROL.json": "5a32f2277bc878f56818f289b81b52c083e3dee95e706b7be99b805242412a68",
+  "census.py": "ccb6d6c4331b5b3a0e3cd0274bb9d0472f6f13c7b6d911a3410ffa6eaa1cc27c",
+  "independent.py": "12db3de009c6a5fca813ead6f06612abb0fb9172fc244b980c289e1756fdeba0",
+  "inputs/U8-WORD.json.gz": "5320f56557816b33e9e708f2134156c72bf8c096de7edaef0169f984e5942686",
+  "verify.py": "d21ce2fe0eff2c748e00765de35cf733d529ce689e255c1dfeb6014a54b57514"
+}

research/u8-terminal-cone-census/PROOF.md

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+# Necessary terminal cones under monotone common-frame restoration
+
+Let `P` be the exact compensated U8 scalar prefix and `B_k` its end-of-B
+binary frame on register k. There are 728 independent arbitrary scalar
+inputs, including 504 arbitrary helper contents. Fix the prefix, its frame
+assignments, entrance frames and end-B frames. The suffix may change its
+scalar word and helper endpoints, subject to these restrictions:
+
+- Only invertible linear ADD and nonzero SCALE primitives on the same registers.
+- Each ADD acts on two registers at the same current frame.
+- After end B, frames only grow by subspace inclusion.
+- No new COPY, ERASE, clean ancilla, frame reset or descent.
+- The prescribed final data shear is independent of incoming helper dirt;
+  each helper is restored identically, without residual scalar helper mixing.
+
+These are the scalar/common-frame hypotheses of this note. Native lowering,
+phases and physical hosting are additional obligations for a construction.
+
+For helper q, every possible terminal frame `U_q` contains
+
+```
+C_q = span(B_q,
+           {B_k : (P^-1)[q,k] != 0},
+           {B_k : k != q and P[k,q] != 0}).
+```
+
+All supports refer to exact rational nonzeros. This joins full subspaces,
+not the count of distinct generators. The criterion follows the cold-ended
+helper lineage in PR591/604/635; this note applies it to all 504 U8 helpers.
+
+The first term follows directly from monotonicity. For the inverse-row term,
+the suffix scalar map is `C = J P^-1`, where J is the prescribed final shear.
+The helper row of J is the q-th identity row, so `C[q,k]=(P^-1)[q,k]`.
+Track formal dependencies on end-B coordinates during the suffix. A nonzero
+dependency originating at register k can reach q only along ADD edges whose
+common frames contain `B_k`; frame growth never removes that subspace.
+Thus every nonzero entry of the final row requires `B_k` contained in `U_q`.
+Cancellation can remove a dependency, but cannot create an entry without
+such a path.
+
+For the dirty-column term fix a proposed `U_q`. At suffix time t let `H(t)`
+be the registers whose current frame is **not** contained in `U_q`. The set
+only grows. Equal-common-frame gates never cross from H to its complement:
+the two incident frames either both lie in `U_q` or both do not. Scalar
+gates within H are invertible on its entire vector of coefficients of the
+independent incoming dirt variable q. Promotions append coordinates to that
+vector, preserving a preexisting nonzero vector. Hence a nonzero q-column
+coefficient in any register initially in H cannot disappear from the entire
+H vector. The final q frame is `U_q`, so q is outside H; every other final
+register has zero q-column coefficient by the prescribed endpoint. The final
+H vector is therefore zero. It follows that initially `P[k,q]=0` for every
+k in H, proving the dirty-column clause.
+
+This promotion argument uses arbitrary independent helper contents and
+invertibility. A cleanup with erasure, extra clean storage, a reset, unequal
+frame ADD or a nonidentity helper endpoint is outside the statement.
+
+The exact finite census finds `C_q = U0_q` for **every** current helper
+endpoint U0. Therefore any terminal frame strictly contained in U0 is
+excluded under these hypotheses. For a dimension-preserving different
+orientation, containing C also forces equality to U0. Larger endpoints
+remain possible. This does not prove that the present tail has minimum
+paid cost: one can seek cheaper climbs or scalar words at the same endpoints.
+Changed prefix dirt responses or end-B frames also escape the result.
+
+The row plus own-monotonicity clause alone closes 496 helpers. Helpers
+347, 349, 359, 361, 370, 371, 372 and 373 have a rank-eight row-only minimum
+and a rank-nine joined minimum. Their independent dirt columns supply the
+missing dimension. All begin in rank one; their residual widths remain
+eight. These eight indices are not identified with U8's eight added holders.
+
+The structural argument above is a written lemma; the programs independently
+certify its finite premises and joins. Neither is a formal verification of
+the all-size integer-multiplication theorem.
+
+Prepared for ToikY with substantial OpenAI Codex assistance. Apache-2.0.

research/u8-terminal-cone-census/README.md

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+# Complete U8 terminal-cone census
+
+**Independent verification / scoped negative result; no new kappa.** For
+the pinned compensated U8 prefix and literal end-B frames, the necessary
+inverse-row, dirty-column and own-monotonicity cone equals the present
+terminal frame for every one of the 504 helpers. No smaller endpoint is
+available under the monotone, equal-common-frame invertible scalar cleanup
+contract in [PROOF.md](PROOF.md).
+
+The criterion is inherited. PR864 already reported this test for all378
+FULL-ending helpers of the immediate X4 ancestor, without packaging it.
+This contribution extends the concrete coverage to the changed U8 prefix,
+all504 helpers including its126 low endpoints, and independently reconstructs
+every support and frame with two different exact implementations. It also
+records the eight false admissions from omitting the dirty-column clause.
+No new cone method or first X4 endpoint obstruction is claimed.
+Adjacent PR974 (`f40cd20a8c257bcfee17508c7de68fee24df0568`) prices favorable
+U8 role/phase deletion budgets at fixed idle hosting; it does not recheck
+cones or construct a changed restoration interface. Its budget screen is
+separate from this complete endpoint census and whole-suffix accounting.
+
+Run from the repository root with Python3.10 or later:
+
+```sh
+python3 -I -B research/u8-terminal-cone-census/verify.py
+```
+
+Expected status is `PASS_EXACT_INDEPENDENT_U8_TERMINAL_CONE_CENSUS`, with
+504helpers,728arbitrary input columns and no smaller-endpoint candidates.
+The package is self-contained: copying this directory to another location
+needs only the Python standard library. `--optimized` also checks the child
+interpreters with optimization; rejection gates do not rely on assertions.
+No optimizer, floating tolerance or supplier auditor is imported.
+
+`CERTIFICATE.json.gz` contains every helper support and canonical frame;
+verification compares its decompressed proof-relevant payload. Container
+hashes bind committed inputs, not regenerated compression streams.
+
+The primary uses chronological sparse `Fraction` rows. The independent
+implementation right-composes integer/dyadic columns in the opposite order
+and enumerates all512 GF(2)^9 vectors to build subspaces. Each independently
+rederives6320 compensation gates, the12913-gate prefix, the exact complete
+arbitrary-dirt endpoint and prefix inverse. All504 records are compared,
+including their exact supports and canonical subspace bases. The independent
+CLI was adapted only to accept additional explanatory primary record fields;
+its mathematical algorithm is unchanged from the separately executed audit.
+
+| Result | Exact census |
+| --- | --- |
+| Current and necessary helper widths | `{4:80,5:32,7:54,8:122,9:216}` |
+| Joined cone strictly below current endpoint | 0of504 |
+| Closed by inverse row plus own monotonicity | 496of504 |
+| Dirty-column clause closes the remaining dimension | helpers347,349,359,361,370,371,372,373 |
+| Complete helper width / idle mass per replica | 3778 /14382 |
+
+The eight exceptional helpers have entrance rank1 and current terminal rank9,
+so their width is8. Their row-only minimum has rank8. They are potential
+targets only for a mechanism that escapes the column invariant, not valid
+rank-seven residual constructions.
+
+The reproducible verifier compares the complete committed certificate,
+tests every proper one-generator endpoint deletion, checks the eight real
+row-only failures, exact width/idle/deficit accounting and four independent
+semantic controls (zero cancellation, inverse-row orientation, dirt columns
+and GF2 joins). It also executes corrupted-input and existing-output CLI
+controls. The separately executed audit in `audit/` preserves six actual
+semantic mutation rejections and a failed Python3.9 attempt; public historical
+receipts replace local paths with portable markers and retain the hashes of
+the complete private originals. They are labeled derivatives, not byte-exact
+original receipts; the originals are not runtime dependencies.
+
+[COST.md](COST.md) gives the complete favorable moment relaxation and exact
+recurrence accounting. It corrects the provisional single-final-climb cost
+screen: old endpoints are not uniformly FULL, and actual suffix climbs may
+be split. This result excludes narrower endpoints. It does not exclude
+cheaper tails at unchanged endpoints, changed prefixes/end-B frames, added
+storage or a different cleanup/compiler interface. There is no numerical
+global optimality, runtime improvement or larger-kappa claim. Native phases,
+wrappers, hosting, all-size/compiler realization, precision, tape and
+analytic transfer remain separate construction obligations.
+
+Source base: `007664306fff84fbb1c9a1b4053b69f439764034`. The byte-pinned U8
+input is the PR869/870 RECIPE-U8 payload
+`090fd574747fbf9628e70cb9beb792732914cfa52e1df147579fe4dbccc68c5a`.
+It is included directly and completely checked as a finite scalar/frame
+word; this command does not regenerate its supplier ancestry or recertify
+the selected construction. [SOURCES.json](SOURCES.json) records context Git
+blobs, recipe, inherited criterion, predecessor and AI attribution.
+
+The sprint's orientation selected PR890; by15:19UTC main
+`bbc600db2fc15007b7c914f4f2b2ed55fa909c87` selected merged PR942 at
+`1661960547713947/10^18`, matching the independently preserved T37 tie.
+This note leaves the selected result and upstream files untouched. Bounded
+overlap inspection through PR971 found no complete U8 census in the inspected
+new bodies; that is not a comprehensive priority claim.
+
+Sprint seed51, recursive-interface track, began2026-10-13T14:49:33.985287Z.
+Its domain was exactly all504 helpers of this fixed word; the stopping rule
+was completion of every exact cone and independent record comparison,
+rather than elapsed-time or solver exhaustion. No seed reroll occurred.
+
+Credits: icekylinx's arbitrary-dirt audit; Jeremy Kothe's PR869/870 U8;
+PR591/604/635 cold-ended helper work; @navinor's PR864 X4 frame audit;
+Chafik Boukhalfa's PR890 composition; Jonathan Capalbo's PR942 fixed-inventory
+bound; and contributors retained by these sources. Prepared for ToikY with
+substantial OpenAI Codex assistance, including an independent agent
+implementation and review. No independent human review is claimed.
+New material is Apache-2.0; the included U8 word retains repository attribution.