Independent verification: quadratic screening brackets sharpen selected D/E diagnostics

research/selected-quadratic-root-brackets/README.md

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+# Quadratic screening brackets for the selected supplier roots
+
+**Independent analytic screening, not endpoint-grid certification or a new κ.**
+For the three frozen selected pricing ledgers, two rational surrogate curves
+bracket the moment root more than **418×** more tightly than the gap left by
+the linear `a < D/E` diagnostic. All modeled surcharges are retained as
+exponential terms. No arbitrary fractional-power evaluation is used to obtain
+the brackets, and no predecessor pricing function is imported.
+
+The selected endpoint signs already have several independent full replays.
+This contribution instead supplies a sharper screening implication and a
+source-bound check of that implication. It does not replace those replays.
+
+## Exact model and comparison theorem
+
+For the normalized bill, write
+
+```text
+M(a) = sum_i c_i exp(lambda_i*a),   c_i>0, lambda_i>=0,
+D    = 1-sum_i c_i > 0,
+E    = sum_i c_i lambda_i > 0,
+F    = sum_i c_i lambda_i^2,
+L    = max_i lambda_i.
+```
+
+An ordinary rank-r child contributes `c_i=n_r*r/(m*W)` and
+`lambda_i=ln(m/r)`. The modeled bad-residue surcharge contributes
+`rho*m^a`, with `rho=32*m*sum_r n_r/(10^16*W)`: its weight is rho and its
+logarithmic rate is ln(m). It contributes to **D, E, F and L**, not just D.
+The physical derivation of this inherited surcharge remains assumed.
+With no surcharge, this normalized D/E equals the repository's raw
+`(mW-sum n_r r)/(sum n_r r ln(m/r))` diagnostic.
+
+For `0<=t<3`,
+
+```text
+1+t+t^2/2 <= exp(t) <= 1+t+t^2/[2(1-t/3)].
+```
+
+The lower inequality follows from the nonnegative exponential series. For
+the upper inequality, `n! >= 2*3^(n-2)` for every n>=2: equality at n=2,3,
+and induction thereafter. Summing that geometric coefficient majorant gives
+the rational upper expression. Applying it to every term, and using
+`lambda_i<=L`, gives
+
+```text
+E*a + F*a^2/2 <= M(a)-M(0)
+             <= E*a + F*a^2/[2(1-L*a/3)].
+```
+
+For certified nonnegative log intervals, let `Elo<=E<=Ehi`,
+`Flo<=F<=Fhi`, and `L<=Lhi`. Define
+
+```text
+Q(a) = Elo*a + Flo*a^2/2,
+U(a) = Ehi*a + Fhi*a^2/[2(1-Lhi*a/3)],
+H    = D/Elo, requiring Elo>0 and Lhi*H<3.
+```
+
+Both Q and U are strictly increasing on [0,H], start at zero and reach at
+least D at H. Their roots at height D are therefore bracketed there.
+The true moment is strictly increasing and unbounded because some positive
+weight has positive rate, so it has exactly one positive root a* at height one.
+The comparison proves
+
+```text
+root(U=D) <= a* <= root(Q=D) <= H.
+```
+
+Exact rational bisection reports the **lower endpoint of the U-root bracket**
+and the **upper endpoint of the Q-root bracket**. The inner endpoints would
+not certify this implication. The implementation verifies the final signs
+and rejects the pole/domain or an unbracketed surrogate.
+
+## Selected evidence
+
+Profiles are read from the selected assembly and bit `nine.profile`, not from
+PR headlines. The complex profile, rank mass and deficit are cross-bound to
+the retained independent owner receipt. The bit rank mass and endpoint are
+also matched, and the selected surcharge flags checked. Exact live-input and
+historical provenance pins are in [SOURCES.json](SOURCES.json).
+
+The following decimals are **outward rounded** displays of the exact rational
+root brackets in [RESULT.json](RESULT.json). Narrowing is
+`(linear_upper-published_endpoint)/(root_upper-root_lower)`; it measures a
+diagnostic uncertainty gap, not a speedup or exponent improvement.
+
+| Model | Certified lower | Certified upper | D/E upper | Narrowing factor |
+|---|---:|---:|---:|---:|
+| Native complex, m=73, W=14336 | 0.00167546136793417 | 0.00167547358407120 | 0.00168059070975433 | >419 |
+| Conservative complex, same profile + surcharge | 0.00167546136507270 | 0.00167547358120967 | 0.00168059070687533 | >419 |
+| Bit, m=64, W=28416, surcharge | 0.00173342514312811 | 0.00173343740334675 | 0.00173855724840137 | >418 |
+
+Every interval contains the published endpoint and its next 10^-24 grid
+point. **Containment does not reprove acceptance or rejection at those points.**
+The approximately 1.2e-8 surrogate gap also cannot resolve the much smaller
+native/conservative correction. Finer bisection alone will not remove this
+analytic approximation gap.
+
+## Reproduce and controls
+
+Python 3.10+, standard library only, from the repository root:
+
+```sh
+python3 -I -O -B research/selected-quadratic-root-brackets/verify.py
+python3 -I -O -B research/selected-quadratic-root-brackets/test_verify.py
+python3 scripts/research_package.py --package research/selected-quadratic-root-brackets --check-only
+python3.12 -B scripts/research_package.py --package research/selected-quadratic-root-brackets
+```
+
+Expected verifier output:
+
+```text
+PASS_SELECTED_QUADRATIC_ROOT_BRACKETS ledgers=3 controls=12 narrowing_gt_100x surcharge_included no_new_kappa
+```
+
+The independently written logarithm primitive uses binary reduction and
+72 positive atanh terms, an explicit geometric tail, and outward 160-bit
+dyadic rounding. On `[1,2]`, z=(x-1)/(x+1) lies in [0,1/3]. After 72 terms,
+the remaining log series is at most `2*z^145/[145*(1-z^2)]`.
+The log method itself is not claimed as new or distinct from every retained
+log implementation: the **moment-root enclosure** is the rational surrogate
+comparison rather than another full exponential-series endpoint replay.
+
+Twelve check groups cover log-one, a log-two decimal interval, a single
+exponential root, an independent exact dyadic root, six invalid-domain or
+unbracketed-root rejections, every surcharge D/E/F contribution and a fixture
+where the surcharge strictly raises the maximum logarithmic rate. The
+exact fixture `M(a)=2^(a-1)` has root 1; broad rational bounds
+`2/3<ln(2)<7/10` give `Q(1)=4/9<1/2<U(1)=469/920`, so it checks the root
+directions without evaluating fractional powers. The separate CLI tests run
+under `-O` and reject altered root, surcharge and source-hash receipts with
+the expected diagnostic. Default replay is read-only; `--write` explicitly
+regenerates RESULT after the mathematical checks. Local registered replay
+is bounded and requires no installations.
+
+## Task and scope
+
+Base `2de86c77b4127d2f01f504e0c7c5e4db737caa3c`; track
+`independent-verification`; card `independent-moment`; catalogue scope key
+`c191650c0fa8d2fa514e`; OS nonce `4fa6ddc953abae43610a0c49ace10c99`.
+[TASK.json](TASK.json) records the refined three-ledger domain, stop rule,
+source heads and local card. The first bounded computation succeeded; no
+pivot was used. The baseline binding command passed before computation.
+
+The entry survey and immediate concrete-object overlap refresh found the
+prior linear diagnostic, its log-contract audit and existing selected
+endpoint audits, not a matching quadratic screening package. This is not
+proof of novelty. Claim:
+https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-18915959.
+No directly matching open handoff was found; the chosen card's independent
+numeric-enclosure route was retained rather than forcing an unrelated task.
+
+No word, frame, ownership, hosting, precision/tape, all-size compiler,
+analytic-transfer or bit H1-H5 obligation is rechecked. No full cold replay,
+supplier adoption, selected-frontier edit or new κ is claimed. Existing
+profile realization and surcharge-contract assumptions remain inherited.
+
+## Open next questions
+
+Can a certified surrogate reject one pinned pending changed profile before
+a full moment comparison? A sharper cheap rejection screen could save
+pricing work, without proving physical feasibility. A feasible first step
+is one exact profile/charge model, checking D>0 and `Lhi*D/Elo<3`, then
+comparing with its independent full replay. The blocker is that this package
+covers only three frozen selected ledgers; pending-profile source binding,
+complete invoice and actual timing benefit must be established separately.
+Structured fields are in [TASK.json](TASK.json).
+
+## Credit and license
+
+Credit the selected construction and reviewed source lineage: Rohan Arun
+(https://github.com/CrocSwap/integer-mult-bounds/pull/1094), Jeremy Kothe
+(https://github.com/CrocSwap/integer-mult-bounds/pull/1087 and mutable-leaver
+lineage), Hao Yan (https://github.com/CrocSwap/integer-mult-bounds/pull/858),
+and DreamingOfClouds' retained bit supplier
+(https://github.com/CrocSwap/integer-mult-bounds/pull/589), with eumemic and
+the full predecessor credit in the selected review. Sources are untouched.
+Aaron Wang / navinor's prior linear diagnostic
+https://github.com/CrocSwap/integer-mult-bounds/pull/1293 motivates the
+comparison; TennisBowling's https://github.com/CrocSwap/integer-mult-bounds/pull/1405
+is credited for the fail-closed log-contract audit. Neither numerical routine
+is imported or reused. Existing complete endpoint replays retain their credit.
+
+Prepared by Aaron Wang / navinor, **with help from chatgpt**, including
+separate read-only AI proof/code review. No independent human or
+proof-assistant review is claimed. Apache-2.0.

research/selected-quadratic-root-brackets/RESULT.json

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+{
+  "sources": {
+    "research/community-frontier1094-audit/ASSEMBLY.json": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045",
+    "research/community-frontier1094-audit/INDEPENDENT1094.json": "301fb8a0e4ab7e3c383ff974c54c08178283b2a3453c5282757f692a3c7a4bfd",
+    "research/e8-bit-p111-tuned/tune/EXPECTED.json": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4"
+  },
+  "ledgers": [
+    {
+      "D": "5/1022",
+      "E": [
+        "2226267329607014812531773732109440320235069757525462461/764751192756317584590932940306453117997241886167793664",
+        "4452534659214029625063547464218880640470139515055948977/1529502385512635169181865880612906235994483772335587328"
+      ],
+      "F": [
+        "423642475897354140413299417883439912745134363922753651212945362074467456741824477438689136542666162239/39917325727289967619798165644896589917695342113801126964770313566981642235388610598829115480670732288",
+        "3389139807178833123306395343067519301961074911389585273831274365789791055719806875074238397406521732929/319338605818319740958385325159172719341562736910409015718162508535853137883108884790632923845365858304"
+      ],
+      "L_upper": "783814187267525543074368603512103785548504437263/182687704666362864775460604089535377456991567872",
+      "surcharge_coefficient": "0",
+      "linear_upper": "3741444191567111470601433171753684530319187310018560/2226267329607014812531773732109440320235069757525462461",
+      "root_lower": "3730024905450536003526167599368321493128347385856000/2226267329607014812531773732109440320235069757525462461",
+      "root_upper": "1243350700612425834451262209817631057636993489960960/742089109869004937510591244036480106745023252508487487",
+      "width": "27196386741499827619030084571679782633084026880/2226267329607014812531773732109440320235069757525462461",
+      "log_bits": 160,
+      "log_terms": 72,
+      "bisection_steps": 100,
+      "name": "native_complex",
+      "m": 73,
+      "W": 14336,
+      "surcharge": false,
+      "endpoint": "67018507072324140917/40000000000000000000000",
+      "rank_mass": 1041408,
+      "narrowing_factor": "456654888532557295881935749100389403969058823851099670637704755042383263/1087855469659993104761203382867191305323361075200000000000000000000000",
+      "narrowing_factor_floor": 419
+    },
+    {
+      "D": "114285714091334443/23360000000000000000",
+      "E": [
+        "695708540510724158908723380275351679746858872586852300332113580615203/238984747736349245184666543845766599374138089427435520000000000000000",
+        "86963567563840519863590422534418959968357359073454663615734138153437/29873093467043655648083317980720824921767261178429440000000000000000"
+      ],
+      "F": [
+        "3706871664155348937851154683532397509559433283033197998212737066209490423583928468326968318347378591103637959795768691/349276600113787216673233949392845161779834243495759860941740243711089369559650342739754760455868907520000000000000000",
+        "57919869752427327153924291930193711086866145047522850677383452094693433773943424726036159606085544544394630728072931/5457446876777925260519280459263205652809910054621247827214691307985771399369536605308668132122951680000000000000000"
+      ],
+      "L_upper": "783814187267525543074368603512103785548504437263/182687704666362864775460604089535377456991567872",
+      "surcharge_coefficient": "18639163/2240000000000000000",
+      "linear_upper": "1169201307876117125825584202814113553498009201483363467256005656576/695708540510724158908723380275351679746858872586852300332113580615203",
+      "root_lower": "1165632780976839579968930177935206036867808848298459602692958322688/695708540510724158908723380275351679746858872586852300332113580615203",
+      "root_upper": "1165641279847652929602261155879619754480749985731745786272635944960/695708540510724158908723380275351679746858872586852300332113580615203",
+      "width": "8498870813349633330977944413717612941137433286183579677622272/695708540510724158908723380275351679746858872586852300332113580615203",
+      "log_bits": 160,
+      "log_terms": 72,
+      "bisection_steps": 100,
+      "name": "conservative_complex",
+      "m": 73,
+      "W": 14336,
+      "surcharge": true,
+      "endpoint": "1675462673946628390003/1000000000000000000000000",
+      "rank_mass": 1041408,
+      "narrowing_factor": "3567616304512985759694051176976209232156984041838810332016144209905296166479833844984391/8498870813349633330977944413717612941137433286183579677622272000000000000000000000000",
+      "narrowing_factor_floor": 419
+    },
+    {
+      "D": "16326904278023/2890625000000000",
+      "E": [
+        "857816062600787715311825029205937383584944311835989118313192937/264040823150602577995782904348156600230808125440000000000000000",
+        "13725057001612603444989200467294998137359108989389858028020965729/4224653170409641247932526469570505603692930007040000000000000000"
+      ],
+      "F": [
+        "34178575185287116286055432031537844231841197666129604174357369819366745552809403008560658509722331673539651550591/3087168762854440353461554809698180790636721379968599706564083573805551947417332044955062391275520000000000000000",
+        "68357150370574232572110864063075688463682395332397527451134424773488254505732667886567689030007893245740993464383/6174337525708880706923109619396361581273442759937199413128167147611103894834664089910124782551040000000000000000"
+      ],
+      "L_upper": "3039107217898977213407095817540395654759610698101/730750818665451459101842416358141509827966271488",
+      "surcharge_coefficient": "4713/722656250000000",
+      "linear_upper": "1491362333429721118405064796109867521564479852090397481238528/857816062600787715311825029205937383584944311835989118313192937",
+      "root_lower": "1486959931091370769655699369190842305347682679541384406368256/857816062600787715311825029205937383584944311835989118313192937",
+      "root_upper": "135179131645803832552914005658087330608515987248426722000896/77983278418253428664711366291448853053176755621453556210290267",
+      "width": "10517012471388426354693048118331345993180191309535641600/857816062600787715311825029205937383584944311835989118313192937",
+      "log_bits": 160,
+      "log_terms": 72,
+      "bisection_steps": 100,
+      "name": "bit",
+      "m": 64,
+      "W": 28416,
+      "surcharge": true,
+      "endpoint": "1733427039589844234749/1000000000000000000000000",
+      "rank_mass": 1808352,
+      "narrowing_factor": "4400775523021170984351065424627457254861917496773458391989867891439947364443232187/10517012471388426354693048118331345993180191309535641600000000000000000000000000",
+      "narrowing_factor_floor": 418
+    }
+  ],
+  "controls": 12,
+  "scope": "screening brackets only; no endpoint-grid certification, word replay or new kappa"
+}

research/selected-quadratic-root-brackets/SOURCES.json

@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "base_commit": "2de86c77b4127d2f01f504e0c7c5e4db737caa3c",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/INDEPENDENT1094.json",
+      "usage": "input",
+      "git_blob": "009a385e26858058a18188364078d162aa6f9519",
+      "sha256": "301fb8a0e4ab7e3c383ff974c54c08178283b2a3453c5282757f692a3c7a4bfd"
+    },
+    {
+      "path": "research/e8-bit-p111-tuned/tune/EXPECTED.json",
+      "usage": "input",
+      "git_blob": "e293ddc557742637f9f679f083a1a282d87b1b13",
+      "sha256": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4"
+    }
+  ]
+}