From 6053794e49861d7c46bb4aee8ef34606633658c9 Mon Sep 17 00:00:00 2001 From: Ali Romman <226877337+aliromman@users.noreply.github.com> Date: Mon, 2 Nov 2026 22:08:32 -0600 Subject: [PATCH 1/5] Prove an all-size P104 caller ceiling for every monotone frame assignment --- .../p104-monotone-frame-ceiling/CASES.json | 40 + .../DERIVATIVE-DUAL-6529.json | 1 + .../DERIVATIVE-DUAL.json | 1 + .../DUAL-6529.json | 1 + research/p104-monotone-frame-ceiling/NOTICE | 16 + research/p104-monotone-frame-ceiling/PROOF.md | 244 +++ .../p104-monotone-frame-ceiling/README.md | 90 ++ .../p104-monotone-frame-ceiling/RESULT.json | 62 + .../SHARP-DUAL.json | 1 + .../p104-monotone-frame-ceiling/SOURCES.json | 76 + .../p104-monotone-frame-ceiling/TASK.json | 123 ++ .../p104-monotone-frame-ceiling/VALIDATION.md | 34 + .../p104-monotone-frame-ceiling/VERIFY.json | 30 + .../p104-monotone-frame-ceiling/discover.py | 70 + research/p104-monotone-frame-ceiling/exact.py | 68 + .../p104-monotone-frame-ceiling/frames.py | 124 ++ .../input/NOTICE-6398 | 287 ++++ .../input/NOTICE-6453 | 18 + .../input/NOTICE-6529 | 21 + .../input/ROOT-PROOF.md | 222 +++ .../input/ROOT.json | 1319 +++++++++++++++++ .../input/WORD-6529.json.gz | Bin 0 -> 87875 bytes .../input/WORD.json.gz | Bin 0 -> 86106 bytes research/p104-monotone-frame-ceiling/model.py | 72 + .../p104-monotone-frame-ceiling/verify.py | 231 +++ 25 files changed, 3151 insertions(+) create mode 100644 research/p104-monotone-frame-ceiling/CASES.json create mode 100644 research/p104-monotone-frame-ceiling/DERIVATIVE-DUAL-6529.json create mode 100644 research/p104-monotone-frame-ceiling/DERIVATIVE-DUAL.json create mode 100644 research/p104-monotone-frame-ceiling/DUAL-6529.json create mode 100644 research/p104-monotone-frame-ceiling/NOTICE create mode 100644 research/p104-monotone-frame-ceiling/PROOF.md create mode 100644 research/p104-monotone-frame-ceiling/README.md create mode 100644 research/p104-monotone-frame-ceiling/RESULT.json create mode 100644 research/p104-monotone-frame-ceiling/SHARP-DUAL.json create mode 100644 research/p104-monotone-frame-ceiling/SOURCES.json create mode 100644 research/p104-monotone-frame-ceiling/TASK.json create mode 100644 research/p104-monotone-frame-ceiling/VALIDATION.md create mode 100644 research/p104-monotone-frame-ceiling/VERIFY.json create mode 100644 research/p104-monotone-frame-ceiling/discover.py create mode 100644 research/p104-monotone-frame-ceiling/exact.py create mode 100644 research/p104-monotone-frame-ceiling/frames.py create mode 100644 research/p104-monotone-frame-ceiling/input/NOTICE-6398 create mode 100644 research/p104-monotone-frame-ceiling/input/NOTICE-6453 create mode 100644 research/p104-monotone-frame-ceiling/input/NOTICE-6529 create mode 100644 research/p104-monotone-frame-ceiling/input/ROOT-PROOF.md create mode 100644 research/p104-monotone-frame-ceiling/input/ROOT.json create mode 100644 research/p104-monotone-frame-ceiling/input/WORD-6529.json.gz create mode 100644 research/p104-monotone-frame-ceiling/input/WORD.json.gz create mode 100644 research/p104-monotone-frame-ceiling/model.py create mode 100644 research/p104-monotone-frame-ceiling/verify.py diff --git a/research/p104-monotone-frame-ceiling/CASES.json b/research/p104-monotone-frame-ceiling/CASES.json new file mode 100644 index 0000000000..c0e40f574e --- /dev/null +++ b/research/p104-monotone-frame-ceiling/CASES.json @@ -0,0 +1,40 @@ +[ + { + "name": "PR6453", + "word": "WORD.json.gz", + "sha256": "e64cc53ea0088191423045cb8563b5bc87a34f6b992baa1593c7917d95a07820", + "kappa": ".0032915", + "source_kappa": ".003287014", + "paid": { + "1": 2236, + "2": 453, + "3": 491, + "4": 109, + "5": 151, + "6": 8, + "7": 9, + "8": 1 + }, + "dual": "SHARP-DUAL.json", + "derivative": "DERIVATIVE-DUAL.json" + }, + { + "name": "PR6529", + "word": "WORD-6529.json.gz", + "sha256": "15ca0c4ee15ad5a58dfc178d7b44d89ca202021c2907fbf9ef2ed50ba1ed8513", + "kappa": ".003293", + "source_kappa": ".003288299", + "paid": { + "1": 2234, + "2": 453, + "3": 491, + "4": 107, + "5": 153, + "6": 8, + "7": 9, + "8": 1 + }, + "dual": "DUAL-6529.json", + "derivative": "DERIVATIVE-DUAL-6529.json" + } +] diff --git a/research/p104-monotone-frame-ceiling/DERIVATIVE-DUAL-6529.json b/research/p104-monotone-frame-ceiling/DERIVATIVE-DUAL-6529.json new file mode 100644 index 0000000000..f51c8b7191 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/DERIVATIVE-DUAL-6529.json @@ -0,0 +1 @@ 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diff --git a/research/p104-monotone-frame-ceiling/NOTICE b/research/p104-monotone-frame-ceiling/NOTICE new file mode 100644 index 0000000000..398b7b55c2 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/NOTICE @@ -0,0 +1,16 @@ +Monotone frame and caller ceilings +New proof, certificate and checker: Ali Romman with substantial OpenAI Codex assistance. +Apache-2.0. No subagents. No independent human or proof-kernel acceptance is claimed. + +PR6529: Ali Romman's scaled-transport word. +PR6453: Erwan L., with the AI assistance stated in its exact source notice. +PR6399 and PR6417: Rohan Arun, with substantial OpenAI Codex assistance. +PR6375: DreamingOfClouds, with the AI assistance stated in the source notice. +PR6398: Ali Romman's mixed root, normalization, child recurrence and all-size tail. + +The original notices are retained unchanged as input/NOTICE-6529, +input/NOTICE-6453 and input/NOTICE-6398. They retain Cinna Davis, David Arias, +Gareth Davidson, Jacob Alan Sussman, zxv, Cao Yifan, Pierre-Loic Doulcet and +all earlier native, helper, BIT, compiler, analytic and manuscript credit. +No predecessor construction is claimed as new. Weak duality, dimension +submodularity and interval series are classical methods. diff --git a/research/p104-monotone-frame-ceiling/PROOF.md b/research/p104-monotone-frame-ceiling/PROOF.md new file mode 100644 index 0000000000..5f8f05eb56 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/PROOF.md @@ -0,0 +1,244 @@ +# Exact monotone-frame cost and caller ceilings + +## 1. Domain and statement + +Use either exact scalar program in `CASES.json`. Both have 770 public fields, +29 owned labels and 104 main ports. The main roles are the live roles r and +r+112. Keep every scalar event, coefficient, COPY, INIT, CLEAR and owned +lifetime. Keep the source main input and output frames. Assign a binary +subspace U of F2^9 to each field segment between MOVE events. At an ADD or +COPY, the two field segments must have the same subspace. Public MOVE events +must go from U to V with U contained in V. Owned MOVE events must go in the +reverse direction. Keep these directions even when the two subspaces become +equal. Each public helper must have nested input and output subspaces. + +Use the source canonical frame maps. A nested move of dimension difference r +costs H(r)=r^(1-kappa). A helper with forward dimension difference r earns H(r). +Set H(0)=0. Every other implementation condition can restrict this domain; +such a restriction cannot weaken a lower bound proved for the larger domain. +The proof does not assert that every assignment in the domain is a physical +native implementation. + +Let D_U(kappa) be the sum of all paid MOVE moments minus all productive helper +moments for an assignment U. The main outputs are priced by the caller. Then: + +1. For PR6453 at kappa=0.0032915, D_U is greater than 1734.45465379. + Its improvement over the original local invoice is less than 0.053864. +2. For PR6529 at kappa=0.003293, D_U is greater than 1734.44478776. + Its improvement over the original local invoice is less than 0.053888. +3. For each word and every U, the derivative of D_U is greater than 3866 + throughout 0<=kappa<=0.0034. +4. The PR6398 mixed root, with every positive integer child size, every binary + child tree and every mixture of these native variants, has negative raw + margin at its corresponding ceiling and above, through kappa=0.0034. + +The domain fixes each source separately. A frame assignment for PR6453 is not +silently applied to PR6529. Both source words and both constraint systems are +rebuilt and checked. + +## 2. Contact components and forced subspaces + +Give each initial field and each new MOVE or INIT segment a distinct node. +Identify the two nodes at every ADD and COPY. This gives 1,063 components +for PR6453 and 1,061 for PR6529. Associate one unknown subspace U_a with each +component. Reverse owned edges, so every directed edge a->b means U_a<=U_b. +The fixed main boundary nodes give fixed subspaces. + +Start with L_a={0} and K_a=F2^9 at unfixed nodes. At fixed nodes set both to the +fixed subspace. Propagate L forward by sums and K backward by intersections +until stable. Every legal assignment satisfies + +```text +L_a <= U_a <= K_a. +``` + +This follows by induction at every propagation step. The original source +assignment is also checked against all resulting bounds. There are 757 and +755 components with different lower and upper bounds. + +For an edge or any directed path a->b, put + +```text +g_ab = dim(L_b) - dim(L_b intersection K_a). +``` + +Since U_a<=U_b, the quotient U_b/U_a contains the image of L_b. Thus +`dim(U_b)-dim(U_a) >= g_ab`. This remains valid when g_ab is zero. +For a pair of incoming edges a->c and b->c, dimension submodularity gives + +```text +dim(U_a)+dim(U_b)-dim(U_c) <= dim(K_a intersection K_b). +``` + +For outgoing edges c->a and c->b, it gives + +```text +dim(U_a)+dim(U_b)-dim(U_c) >= dim(L_a+L_b). +``` + +These statements use the actual binary subspaces. Dimensions alone would not +supply the right-hand sides. The checker computes each sum and intersection +as a set of the at most 512 binary vectors. + +## 3. An exact relaxation of the complete local invoice + +For each component a and each integer d between dim(L_a) and dim(K_a), use +one variable x_(a,d). Their sum at a is one. For each paid or helper edge, +use pair variables x_(a,b,d,e), with `e-d>=g_ab`. Their two marginals equal +the corresponding node variables. All variables are between zero and one. +Encode the path and fork inequalities of Section 2 as linear inequalities +in the expected dimensions. + +A legal subspace assignment gives a zero-one feasible point: choose its exact +dimension at each node and its exact dimension pair at each edge. We do not +need the converse. Fractional points or unrealizable dimensions only enlarge +the feasible set. + +Combine equal edges with signed integer weights. A paid MOVE contributes +1; +a public helper from its input component to its output component contributes +-1. Its tariff is + +```text +weight * H(e-d). +``` + +A zero-weight edge can retain constraints. A zero-rank edge has zero tariff. +The public helper credit is not frozen at its source value. The complete +source controls are: + +| Rank | PR6453 paid | PR6529 paid | Helpers in each source | +| --- | ---: | ---: | ---: | +| 1 | 2236 | 2234 | 14 | +| 2 | 453 | 453 | 8 | +| 3 | 491 | 491 | 4 | +| 4 | 109 | 107 | 107 | +| 5 | 151 | 153 | 15 | +| 6 | 8 | 8 | 3 | +| 7 | 9 | 9 | 9 | +| 8 | 1 | 1 | 39 | +| 9 | 0 | 0 | 363 | + +Each paid mass is 5925 and each helper mass is 4205. Their difference is +1720. Owned reverse moves have total rank 56. These controls do not replace +the variable helper debit in the proof model. + +The two exact models have respectively 13,266 and 13,132 variables, 12,793 +and 12,697 equalities, and 119,534 and 119,463 inequalities. Their canonical +integer descriptions are bound by SHA256 in each dual certificate. Model +construction uses no solver. + +## 4. Rational weak-duality certificates + +Write the exact relaxation as Ax=b, Bx<=d, 0<=x<=1. For any rational y and +any componentwise nonpositive rational z, + +```text +c*x >= b*y+d*z + sum_j min(0, c_j-(A^T*y+B^T*z)_j). +``` + +Indeed, `y*A*x=y*b`, `z*B*x>=z*d`, and each residual coefficient times +x_j is at least its minimum with zero. Thus a rounded dual does not need an +exact zero residual. The residual correction is included in full. + +The four retained duals use integer numerators and denominator 10^10. The +checker validates each index, disallows duplicate indices, checks each upper +bound dual sign, reconstructs every residual and sums its negative part. +No numerical optimizer status is used as proof. + +`exact.py` encloses every needed H(r) with integer intervals in units of +2^-160. To enclose log(r), reduce to a ratio in [1,2] and use + +```text +log(a/b) = 2 sum_(j>=0) z^(2j+1)/(2j+1), z=(a-b)/(a+b). +``` + +Here 0<=z<=1/3. After 80 terms, the remaining sum is at most +`2*z^161/(161*(1-z^2))`. Every division rounds outward. The exponential +uses terms through degree 48 for an argument in [0,1]. Twice the outward +upper bound on term 49 bounds its tail. Inverting the resulting interval +for exp(kappa*log(r)) gives the H(r) interval. The replay checks the argument +bounds before each use. No floating value decides a proof condition. + +For the derivative, a tariff w*H(r) has derivative -w*H(r)*log(r). If w>=0, +a uniform lower bound on [0,.0034] is -w*r*log(r). If w<0, a uniform lower +bound is -w*H_.0034(r)*log(r). Use outward endpoints for both expressions. +A separate rational dual for each word proves the lower bounds +3866.27671714 and 3877.25840291. Thus every fixed legal assignment has a +strictly increasing native deficit on this interval. The model itself does +not depend on kappa. + +## 5. Every binary child and the complete fixed root + +The root and recurrence are from PR6398 Section 5, retained byte for byte in +`input/ROOT-PROOF.md`. The source root has outputs with stage supports +ABCD, AB, CD, empty, C and B. It pays three thin queries and individual +whole idles for B and C. Every full site contains a binary query supplier. +A binary node with n leaves pays two whole idles H(8n), one thin query H(1), +and earns two independent main targets H(8n+1). + +Let L be the certified native lower bound divided by 104. Let C(n) be the +minimum binary-node cost for n leaves. Then + +```text +C(1)=0, +C(n)=min_(1<=a=100, concavity gives `C(n)>=-(n-1)` and +`H(8b+1)-H(8b)<=H(9)-H(8)`. Jensen's inequality gives + +```text +B(N)=H(8N+1)+2H(4N+1)-N(L-1)+2(H(9)-H(8))-6. +B'(N)=8(1-kappa)*[(8N+1)^(-kappa)+(4N+1)^(-kappa)]-(L-1). +``` + +For both sources B(100)<-1.86 and B'(100)<-0.059. Also B''(N)<0. +Thus B(N)<0 for every real N>=100. No larger tree is omitted. + +Each positive H term decreases with kappa. Each increment H(r+1)-H(r) +for r>=8 decreases, because x^(1-kappa)*log(x) increases for x>=1. +Every binary-node cost consequently increases. Section 4 proves the same +property for every native deficit. Therefore each complete caller margin +decreases through kappa=.0034. The exclusions extend from their tested +ceilings to that cap, even if one chooses a different tree or frame +assignment at each kappa. + +## 6. Interpretation and limits + +The latest source comparison is PR6529's reported conditional .003288299. +The ratio `.003293/.003288299-1` is less than .00143. Thus the whole stated +frame-only class has less than .143 percent remaining headroom. This is an +upper bound on possible improvement, not an attained construction. + +The proof does not cover scalar-word changes, added contacts, nonnested +Clifford frames, altered main boundaries, transported noncommuting codes, +a different root or a different binary module. It does not establish an +optimum for all P104 suppliers. Native phase correctness, physical ownership, +precision, compiler and outer admission of the source constructions are +retained premises; their full cold replays are not repeated here. Granting +all extra nonnegative costs zero is valid for a negative raw-margin result. +It would be invalid for a positive kappa claim. + +Weak duality, the dimension identities and the series bounds are standard. +The new evidence is the source-specific joint constraint system and its +uniform all-size consequence with changed helper credits. All checks in this +package are author-side and AI-assisted. The donor's requested large complete +kappa improvement has not been achieved. diff --git a/research/p104-monotone-frame-ceiling/README.md b/research/p104-monotone-frame-ceiling/README.md new file mode 100644 index 0000000000..04a5678b30 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/README.md @@ -0,0 +1,90 @@ +# A ceiling for every monotone frame assignment on two P104 words + +**No new kappa is proved.** This package excludes a large gain from a whole +class of frame changes, including joint changes that a local search can miss. +For the exact PR6529 scalar word, every allowed assignment fails at +**kappa 0.003293** and at every larger value through 0.0034. The fixed mixed +root can use every positive child size, every binary tree, and a different +allowed native variant at each leaf. Thus this class has less than +**0.143 percent** headroom over PR6529's reported 0.003288299. +At the 0.003293 ceiling, different leaves can also use different source words. + +The proof keeps the scalar events, contact equalities, public forward moves, +owned reverse moves and main boundaries. It permits all binary subspaces at +all contacts, including new subspaces absent from the source. Helper input +and output frames can change. Every paid rank and changed helper credit is +charged. A rank-zero helper has zero credit. The proof also covers PR6453, +with ceiling 0.0032915. These ceilings do not apply to a changed scalar word, +nonnested Clifford frames, different main boundaries or a different caller. + +| Exact source | Native saving upper bound at its ceiling | Caller ceiling | +| --- | ---: | ---: | +| PR6453 | less than 0.053864 | 0.0032915 | +| PR6529 | less than 0.053888 | 0.003293 | + +The saving is the paid-minus-helper moment per native call, before division +by 104. It is not a kappa increment. The full raw caller margin is negative +at each ceiling. Ordinary work, routing, precision, compiler and outer fees +are granted zero cost in this exclusion. Adding their nonnegative costs +cannot repair the failure. This is a necessary-budget proof, not a new +supplier or a replay of the complete multiplication theorem. + +`PROOF.md` gives the domain, its reduction to exact constraints, the rational +dual bound and the all-size tail. `RESULT.json` contains exact fractions. +The two models have 119,534 and 119,463 inequalities. Their original words +satisfy every constraint and reproduce the complete signed local invoices. +The original 7,2,2,7 callers pass their published raw prices as controls. +Four rational certificates supply the price and uniform derivative bounds. +A solver is not required for replay. + +## Reproduction + +Use Python 3.12 and the standard library: + +```sh +python3 -I -B research/p104-monotone-frame-ceiling/verify.py +python3 -I -O -B research/p104-monotone-frame-ceiling/verify.py +python3 -B scripts/research_package.py --package research/p104-monotone-frame-ceiling +``` + +The default replay is read-only. It authenticates every retained predecessor, +rebuilds both models, checks all dual coefficients and residuals, and compares +its full result with the retained receipt. `--write` replaces that receipt +with a fresh result. `SOURCES.json` records exact original heads, paths, Git +blobs and SHA256 values. The retained bytes permit offline replay without +pending branch objects or a Git checkout. + +`VALIDATION.md` records completed checks and their limits. Local exact replay +is author-side evidence. Independent scientific review, formal proof-kernel +acceptance and the five framework assumptions remain separate obligations. +The selected frontier is unchanged. + +## Open next questions + +- Change the scalar word or its contact graph. Identify a precise chronological + value relation that removes paid work. Prove every public endpoint and charge + each target loss before using the new invoice. +- Change the frame contract or caller. Nonnested Clifford coframes and different + query roots are outside this theorem. Check current claims first, then prove + one complete endpoint and price it. The structured questions are in `TASK.json`. + +## Credit and coordination + +New proof, exact certificate and checker: Ali Romman with substantial OpenAI +Codex assistance; no subagents. Weak duality, dimension inequalities and +interval series are classical methods. The source-specific combined model, +uniform bound over all frame assignments and leaf mixtures, and complete +all-size caller consequence are this contribution. + +PR6529 supplies the scaled transport word. Erwan L. supplies PR6453's third +anchor; Rohan Arun supplies the PR6399 merge and PR6417 composition; +DreamingOfClouds supplies the shared-carrier lifts. PR6398 supplies the mixed +root, normalization, binary-tree recurrence and concave tail. The earlier +P104, Gaussian, helper, BIT, compiler and analytic authors retain their credit. +Their source notices are copied unchanged under `input/`. See `NOTICE`. + +The canonical investigation is [Discussion5593, comment19362959](https://github.com/CrocSwap/integer-mult-bounds/discussions/5593#discussioncomment-19362959). +Main, priority 8 and selected6030 were checked before work and at each pivot. +The general coframe, period-six caller, source-word and intermediate-retirement +claims were kept separate. The overlap snapshot cannot cover unpublished work. +The user's requested large complete kappa improvement remains unmet. diff --git a/research/p104-monotone-frame-ceiling/RESULT.json b/research/p104-monotone-frame-ceiling/RESULT.json new file mode 100644 index 0000000000..bb0d56e457 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/RESULT.json @@ -0,0 +1,62 @@ +{ + "status": "PASS_EXACT_MONOTONE_FRAME_CALLER_EXCLUSION", + "mixed_source_ceiling": "3293/1000000", + "cases": [ + { + "source": "PR6453", + "kappa": "6583/2000000", + "source_sha256": "e64cc53ea0088191423045cb8563b5bc87a34f6b992baa1593c7917d95a07820", + "components": 1063, + "flexible_components": 757, + "variables": 13266, + "equalities": 12793, + "inequalities": 119534, + "model_sha256": "82099b6b6a185ddb45a2e1911c134c1978a1549978d990db180afcb9f3ec505d", + "native_cost_lower": "12377482013651851476172013105076213923467782108731947160823/7136238463529799405291429847247475681913733120000000000", + "native_saving_upper": "384381180409380114714046155827283855435808608384870427/7136238463529799405291429847247475681913733120000000000", + "dual_correction": "-28304726518988575499789745753277697072052909839288587447049/7136238463529799405291429847247475681913733120000000000", + "native_derivative_lower": "55181345239138230073419447198610096881372259935127859079449/14272476927059598810582859694494951363827466240000000000", + "saving_cap": "17/5000", + "positive_caller_margin_lower": "932156407892285723850992529971769014775133207/75998085141206951746591611301246717022108492234752", + "negative_controls": 7, + "caller": { + "binary_splits": 4851, + "root_pair_checks": 4656, + "finite_margin_upper": "-212460618193477925728567813427453336616344190942416157/371084400103549569075154352056868735459514122240000000000", + "finite_total": 18, + "finite_first_pair": 9, + "tail_from": 100, + "tail_margin_upper": "-1732777769503629082810031740523218871116280529775475181/927711000258873922687885880142171838648785305600000000", + "tail_derivative_upper": "-393602945328301203072468938485329960929772370450294208441293/6621815577647790285361591835278794149907299754311680000000000" + } + }, + { + "source": "PR6529", + "kappa": "3293/1000000", + "source_sha256": "15ca0c4ee15ad5a58dfc178d7b44d89ca202021c2907fbf9ef2ed50ba1ed8513", + "components": 1061, + "flexible_components": 755, + "variables": 13132, + "equalities": 12697, + "inequalities": 119463, + "model_sha256": "6cff339397c7acd70d88bf56b427442201f861020d6c05126083d5fc733e4abc", + "native_cost_lower": "4950964642937357145233505188484714489483760677513434231911/2854495385411919762116571938898990272765493248000000000", + "native_saving_upper": "38455643082472961548509740922119426550410978137047491/713623846352979940529142984724747568191373312000000000", + "dual_correction": 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diff --git a/research/p104-monotone-frame-ceiling/SOURCES.json b/research/p104-monotone-frame-ceiling/SOURCES.json new file mode 100644 index 0000000000..7191307c24 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/SOURCES.json @@ -0,0 +1,76 @@ +{ + "schema_version": 1, + "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a", + "files": [ + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec", + "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81" + }, + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a", + "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa" + } + ], + "retained_predecessors": [ + { + "head": "b1a3e525793eca35c56324aa7ffd0b714cf606d4", + "path": "research/p104-third-anchor-clean-mixed-root/WORD.json.gz", + "git_blob": "08d26d114ec85f8f1d4fd8c96c7d983a2e51a999", + "sha256": "e64cc53ea0088191423045cb8563b5bc87a34f6b992baa1593c7917d95a07820", + "retained_path": "input/WORD.json.gz", + "usage": "Exact external source bytes. The offline verifier checks SHA256 before use." + }, + { + "head": "b1a3e525793eca35c56324aa7ffd0b714cf606d4", + "path": "research/p104-third-anchor-clean-mixed-root/NOTICE", + "git_blob": "28e07986259db6119adcfc8aab3f168c90d7a309", + "sha256": "134f5844514ea3659bf81a844f543713370fd32b65649ee7356b69a2f421e2d5", + "retained_path": "input/NOTICE-6453", + "usage": "Exact external source bytes. The offline verifier checks SHA256 before use." + }, + { + "head": "44b0d0e8961201ac1656490b4384a7752e1cd9c5", + "path": "research/p104-scaled-transport-fusion/WORD.json.gz", + "git_blob": "91baa371a7e5f40c7032a82fc4a64ee163601f40", + "sha256": "15ca0c4ee15ad5a58dfc178d7b44d89ca202021c2907fbf9ef2ed50ba1ed8513", + "retained_path": "input/WORD-6529.json.gz", + "usage": "Exact external source bytes. The offline verifier checks SHA256 before use." + }, + { + "head": "44b0d0e8961201ac1656490b4384a7752e1cd9c5", + "path": "research/p104-scaled-transport-fusion/NOTICE", + "git_blob": "3d9b7e4c8b3a4b34cc65d00895940eea3b3a10b3", + "sha256": "e1b36824e56639f42857d8799d5bfe581616d2bcd402b89be0ae8dda642caaa9", + "retained_path": "input/NOTICE-6529", + "usage": "Exact external source bytes. The offline verifier checks SHA256 before use." + }, + { + "head": "dee2663ba0918be86993fd9d63c8d64b020f4c83", + "path": "research/mixed-projector-query-root/PROOF.md", + "git_blob": "510bf9dba012949ea0f37d9e087ce8778b1951d0", + "sha256": "bfbe0b872f6ff14c858bff47c6e3ac0bb8e43bc8eeaec20253744e3f44720fcd", + "retained_path": "input/ROOT-PROOF.md", + "usage": "Exact external source bytes. The offline verifier checks SHA256 before use." + }, + { + "head": "dee2663ba0918be86993fd9d63c8d64b020f4c83", + "path": "research/mixed-projector-query-root/ROOT.json", + "git_blob": "b59f1ea89aca17f0a76131743c12f2c724421df8", + "sha256": "2f25684c24ae0e1d10d4f8f29430aed3cef584f82c8ae840e82be0a72baa048d", + "retained_path": "input/ROOT.json", + "usage": "Exact external source bytes. The offline verifier checks SHA256 before use." + }, + { + "head": "dee2663ba0918be86993fd9d63c8d64b020f4c83", + "path": "research/mixed-projector-query-root/NOTICE", + "git_blob": "03e4ac5937424ed9f0ae0e4b33d6b0df0c9ec8d9", + "sha256": "b6026dd901f3cb2e6ec6d5f7b32a38aa998005c8a1df93392151ca1766d45b33", + "retained_path": "input/NOTICE-6398", + "usage": "Exact external source bytes. The offline verifier checks SHA256 before use." + } + ] +} diff --git a/research/p104-monotone-frame-ceiling/TASK.json b/research/p104-monotone-frame-ceiling/TASK.json new file mode 100644 index 0000000000..87f7bebad7 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/TASK.json @@ -0,0 +1,123 @@ +{ + "schema_version": 1, + "task_id": "p104-side-correction", + "priority_revision": 8, + "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a", + "original_card": { + "id": "p104-side-correction", + "question": "Can joint helper bases and a legal causal split improve the complete P104 productive/paid invoice?", + "starting_points": [ + "docs/research/current-priorities.md", + "docs/research/research-family-status.md", + "certificates/selected-result.json", + "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md", + "research/p104-helper-subspace-splits/README.md", + "research/dyadic-helper-channel-capacity/README.md" + ], + "methods": [ + "Construct and independently check one source-bound instance of the stated first experiment.", + "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change." + ], + "acceptance_gate": "An exact arbitrary-input identity, source/target frame availability and restoration at the chosen cut, plus the complete channel+correction+idle/hosting bill or an explicit favorable bound. The native channel alone is not a supplier. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.", + "first_step": { + "command": [ + "python3", + "-B", + "scripts/research_selected_baseline.py" + ], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.", + "scope": "Source-bound orientation only, not a supplier replay or theorem proof." + }, + "first_experiment": "Start with the selected #5599 emitted word and review pending #5718 distinct input/output bases plus #5717/#5719 coverage before claiming work. Pick one uncovered source word, helper group and cut region. Prove the arbitrary-input relation and legal entry/exit corrections, then compare full target and paid profiles. Reject if completion, owned lifetimes or consumer costs repay the gain. A static pairing rank alone is not a physical split.", + "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.", + "allowed_variations": [ + "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes." + ], + "scope_change_triggers": [ + "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.", + "Treating an archived proposal or local numerical pass as complete supplier admission." + ], + "overlap_terms": [ + "P104", + "helper basis", + "helper split", + "5718", + "dyadic capacity" + ], + "existing_checks": [ + { + "scope": "PR #4133: literal-slot defect", + "evidence": "research/p104-literal-slot-defect/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "88666f1dee93fbd85b31fd5d6f91a219e111b5f4", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "PR #4136: uniform commutation gap", + "evidence": "research/p104-uniform-commutation-gap/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "281ce601a8be9db470de630759179c23e1dfaa88", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "PR #4140: uniform encoding gap", + "evidence": "research/p104-uniform-encoding-gap/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "de25c6525aa866962309df5d107b32d55e6e5628", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "PR #4153: source-chart shortcut exhausts its optimistic allowance", + "evidence": "research/p104-source-chart-shortcut/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "9123241fea53df77e51ffe30d9aca1625d4bf3a8", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "PR #4154: fiber-tag seam mismatch", + "evidence": "research/p104-fiber-tag-seam/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "4ffcdaee5d6255a9cea1fe53a5424435068089d0", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.", + "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md", + "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.", + "status": "reviewed", + "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137", + "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5" + } + ] + }, + "scope_status": "changed", + "actual_scope": "Prove an exact all-size caller ceiling for all simultaneous monotone contact subspaces on two fixed P104 scalar words, with variable helper boundaries and different variants at different native leaves.", + "scope_change_reason": "The user requires a large complete kappa gain. Bounded frame searches did not give one. This proof decides whether the entire changed-frame contract can reach that gain before another construction search. The current PR6529 source is included after the live refresh.", + "bottleneck_link": "The signed local invoice includes every paid MOVE and changed productive helper. The fixed mixed root and every binary child size are then priced. This excludes frame-only optimization as a route to a large gain in the stated model.", + "budget_minutes": 360, + "stop_rule": "Retain the exact scoped certificate and its controls. Do not call the negative result a kappa improvement or completion of the donor objective.", + "outcome": "negative-result", + "evidence": "PROOF.md, RESULT.json, four exact dual certificates, two source words and verify.py.", + "author": "Ali Romman with substantial OpenAI Codex assistance", + "ai_assistance": "Substantial OpenAI Codex assistance for research, code, proof, verification and prose. No subagents or independent human review.", + "open_questions": [ + { + "question": "Can a changed scalar word reduce the native deficit beyond this frame-only ceiling?", + "why_it_matters": "The current word admits less than 0.055 additional native saving in this contract. A large caller gain needs a different word or interface.", + "first_step": "Name a changed contact, chronological value relation or native map. Prove every arbitrary-input endpoint and charge all target losses.", + "blocked_by": "No affordable new chronological construction is supplied here." + }, + { + "question": "Can nonnested Clifford coframes or a different query root improve the complete price?", + "why_it_matters": "Both changes are outside the theorem. The subspace cuts cannot be applied to them without a new rank contract.", + "first_step": "Check live claims, then test one exact endpoint and its full signed invoice before expansion.", + "blocked_by": "A physical source-bound construction and complete compiler and outer pricing are still required." + } + ] +} diff --git a/research/p104-monotone-frame-ceiling/VALIDATION.md b/research/p104-monotone-frame-ceiling/VALIDATION.md new file mode 100644 index 0000000000..f0a5fb249b --- /dev/null +++ b/research/p104-monotone-frame-ceiling/VALIDATION.md @@ -0,0 +1,34 @@ +# Validation record + +Base: `7c49fe32c55de02c2c873655c804b16e4c3fef0a`. +Current source: PR6529 at `44b0d0e8961201ac1656490b4384a7752e1cd9c5`. +The live PR head and body were checked on 2026-11-03. Priority 8 and selected +PR6030 were unchanged. The source claim is a comparison, not independent +scientific acceptance. Selected binding orientation passed earlier in this +session; the selected cold replay was not run. + +The exact direct normal and optimized replays passed under CPython 3.15.2. +The registered normal and optimized replays passed under CPython 3.12.13 on +macOS arm64, using only the standard library. They check both source systems, +the original signed invoices, fourteen rejection controls, both positive +source callers, every finite caller case and both analytic tails. + +Five separate optimized-process corruptions were rejected: one byte in each +source word, the retained root proof, a false result status and a positive +upper-inequality dual coefficient. These controls used a temporary package +copy. The working source bytes were not changed. + +The source models and four duals were discovered with NumPy 2.5.4 and SciPy +1.18.1 under CPython 3.12.13. Those packages and optimizer status are not used +by the exact verifier. `discover.py` is an optional numerical helper; its +output remains provisional until the exact verifier checks it. + +Metadata and exact source pins passed. Publication-byte, official-export and +sealed replay results will be added after the scientific files are committed. +The checkout is shallow. Full-history ancestry is not claimed. Every pending +predecessor byte required by the proof is retained and checked directly. + +No Lean, Docker, full native supplier, physical phase, rounded multiplier or +selected-frontier cold replay was run for this negative budget result. +Independent human and proof-kernel acceptance remain open. Hosted CI is a +separate status. No kappa gain or completion of the donor objective is claimed. diff --git a/research/p104-monotone-frame-ceiling/VERIFY.json b/research/p104-monotone-frame-ceiling/VERIFY.json new file mode 100644 index 0000000000..3d1a880cf7 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/VERIFY.json @@ -0,0 +1,30 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 240, + "checks": "Rebuild both exact models; check source bytes, all original constraints and complete signed invoices; verify rational dual bounds, derivative bounds, positive callers and fourteen rejection controls; check all finite root sizes and both all-size tails." + }, + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 240, + "checks": "Repeat the exact proof with Python optimization. No assertion decides a proof condition." + } + ] +} diff --git a/research/p104-monotone-frame-ceiling/discover.py b/research/p104-monotone-frame-ceiling/discover.py new file mode 100644 index 0000000000..66ff41a562 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/discover.py @@ -0,0 +1,70 @@ +"""Find a dual candidate. This optional numerical search is not a proof. + +The recorded discovery used NumPy 2.5.4 and SciPy 1.18.1. +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +""" +import argparse +import gzip +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +from frames import graph +from model import model +from verify import model_hash, source_bytes + + +def run(args): + from scipy.optimize import linprog + from scipy.sparse import coo_matrix + from math import log + + spec = json.loads((HERE/'CASES.json').read_text())[args.case] + raw = source_bytes(HERE/'input'/spec['word'], spec['sha256']) + word = json.loads(gzip.decompress(raw)) + variables, cost, eq, ub, tariffs = model(word, graph(word), float(spec['kappa'])) + if args.derivative: + cost = [0.]*len(variables) + for j, weight, rank in tariffs: + cost[j] = (0. if not rank else -weight*rank*log(rank) if weight >= 0 + else -weight*rank**(1-.0034)*log(rank)) + + def sparse(rows): + rr, cc, dd = [], [], [] + for i, (row, _) in enumerate(rows): + for j, value in row.items(): + rr.append(i) + cc.append(j) + dd.append(value) + return coo_matrix((dd, (rr, cc)), shape=(len(rows), len(cost))).tocsr() + + result = linprog(cost, A_ub=sparse(ub), b_ub=[b for _, b in ub], + A_eq=sparse(eq), b_eq=[b for _, b in eq], bounds=(0, 1), + method='highs', options={'time_limit':60}) + if not result.success: + raise RuntimeError(result.message) + scale = 10**10 + certificate = {'model_sha256':model_hash(variables, eq, ub, tariffs), 'scale':scale} + for key, values in [('eq', result.eqlin.marginals), ('ub', result.ineqlin.marginals)]: + rows = [] + for i, value in enumerate(values): + coefficient = round(float(value)*scale) + if key == 'ub': + coefficient = min(0, coefficient) + if coefficient: + rows.append([i, coefficient]) + certificate[key] = rows + with args.output.open('x') as stream: + json.dump(certificate, stream, indent=2) + stream.write('\n') + print('DISCOVERY_ONLY', spec['name'], result.fun, args.output) + + +if __name__ == '__main__': + parser = argparse.ArgumentParser() + parser.add_argument('--case', type=int, choices=[0, 1], required=True) + parser.add_argument('--derivative', action='store_true') + parser.add_argument('--output', type=Path, required=True) + run(parser.parse_args()) diff --git a/research/p104-monotone-frame-ceiling/exact.py b/research/p104-monotone-frame-ceiling/exact.py new file mode 100644 index 0000000000..dbfedab29f --- /dev/null +++ b/research/p104-monotone-frame-ceiling/exact.py @@ -0,0 +1,68 @@ +"""Enclose rank powers with integer arithmetic and explicit series tails. + +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +""" +from fractions import Fraction +from functools import cache + +BITS = 160 +UNIT = 1 << BITS + + +def ceil_div(a, b): + return -(-a // b) + + +def log_unit(a, b): + """Enclose log(a/b), for 1 <= a/b <= 2, in units of 2^-BITS.""" + if not b <= a <= 2*b: + raise ValueError('The logarithm argument is outside [1,2]') + z = Fraction(a-b, a+b) + power = z + lower = upper = 0 + for j in range(80): + term = 2*power/(2*j+1) + lower += term.numerator*UNIT // term.denominator + upper += ceil_div(term.numerator*UNIT, term.denominator) + power *= z*z + tail = 2*power/(161*(1-z*z)) + upper += ceil_div(tail.numerator*UNIT, tail.denominator) + return lower, upper + + +@cache +def log_rank(r): + if not isinstance(r, int) or r < 1: + raise ValueError('The rank must be a positive integer') + b = r.bit_length()-1 + lo, hi = log_unit(r, 1 << b) + two_lo, two_hi = log_unit(2, 1) + return lo+b*two_lo, hi+b*two_hi + + +def exp_small(lo, hi): + """Enclose exp(x), where 0 <= lo/UNIT <= x <= hi/UNIT <= 1.""" + if not 0 <= lo <= hi <= UNIT: + raise ValueError('The exponential argument is outside [0,1]') + lower = upper = term_lo = term_hi = UNIT + for j in range(1, 49): + term_lo = term_lo*lo // (UNIT*j) + term_hi = ceil_div(term_hi*hi, UNIT*j) + lower += term_lo + upper += term_hi + next_hi = ceil_div(term_hi*hi, UNIT*49) + return lower, upper+2*next_hi + + +@cache +def power(r, kappa): + """Return integer bounds for r^(1-kappa) times UNIT.""" + if r == 0: + return 0, 0 + if not 0 <= kappa < 1: + raise ValueError('The saving must be in [0,1)') + lo, hi = log_rank(r) + xlo = lo*kappa.numerator // kappa.denominator + xhi = ceil_div(hi*kappa.numerator, kappa.denominator) + elo, ehi = exp_small(xlo, xhi) + return r*UNIT*UNIT//ehi, ceil_div(r*UNIT*UNIT, elo) diff --git a/research/p104-monotone-frame-ceiling/frames.py b/research/p104-monotone-frame-ceiling/frames.py new file mode 100644 index 0000000000..cd9889d784 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/frames.py @@ -0,0 +1,124 @@ +"""Build exact contact components, subspace bounds and rank invoices. + +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +""" +from collections import Counter, defaultdict, deque + + +def span(basis): + out = {0} + for x in basis: + out |= {y ^ x for y in out} + return frozenset(out) + + +def join(a, b): + return frozenset(x ^ y for x in a for y in b) + + +def dimension(s): + return len(s).bit_length()-1 + + +def graph(w): + parent, old = [], [] + + def node(f): + i = len(parent) + parent.append(i) + old.append(f) + return i + + def root(i): + while parent[i] != i: + parent[i] = parent[parent[i]] + i = parent[i] + return i + + def unite(a, b): + a, b = root(a), root(b) + if old[a] != old[b]: + raise ValueError('The source has an invalid scalar contact') + parent[b] = a + + starts = [node(f) for f in w['initial']] + current = starts+[None]*w['owned_fields'] + moves = [] + segments = [] + for i, e in enumerate(w['program']): + op, r = e[:2] + if op == 'INIT': + current[r] = node(e[2]) + elif op == 'CLEAR': + current[r] = None + elif op == 'MOVE': + a = current[r] + b = node(e[3]) + moves.append((a, b, bool(e[4]), i)) + current[r] = b + elif op in ('ADD', 'COPY'): + unite(current[r], current[e[2]]) + segments.append(current[r]) + ends = current[:w['public']] + roots = sorted({root(i) for i in range(len(parent))}) + index = {r:i for i,r in enumerate(roots)} + label = lambda n: index[root(n)] + return { + 'old': [old[r] for r in roots], + 'starts': list(map(label, starts)), + 'ends': list(map(label, ends)), + 'moves': [(label(a),label(b),p,i) for a,b,p,i in moves], + 'segments': [None if n is None else label(n) for n in segments], + } + + +def bounds(w, g): + full = frozenset(range(512)) + lo = [frozenset({0}) for _ in g['old']] + hi = [full for _ in g['old']] + forward, backward = defaultdict(set), defaultdict(set) + for a,b,private,_ in g['moves']: + if private: + a,b = b,a + forward[a].add(b) + backward[b].add(a) + main = set(w['live']) | {r+112 for r in w['live']} + fixed = set() + for r in main: + for key, field in [('starts','initial'),('ends','final')]: + i = g[key][r] + s = span(w['frames'][w[field][r]]) + if i in fixed and lo[i] != s: + raise ValueError('Inconsistent main boundary') + lo[i] = hi[i] = s + fixed.add(i) + queue = deque(fixed) + while queue: + a = queue.popleft() + for b in forward[a]: + if not lo[a] <= lo[b]: + lo[b] = join(lo[b], lo[a]) + queue.append(b) + queue = deque(fixed) + while queue: + b = queue.popleft() + for a in backward[b]: + if not hi[a] <= hi[b]: + hi[a] &= hi[b] + queue.append(a) + for i, f in enumerate(g['old']): + if not lo[i] <= span(w['frames'][f]) <= hi[i]: + raise ValueError('The source is outside the exact subspace bounds') + return lo,hi + + +def invoice(w,g,spaces): + dims = list(map(dimension, spaces)) + paid = Counter(abs(dims[a]-dims[b]) for a,b,_,_ in g['moves']) + main = set(w['live']) | {r+112 for r in w['live']} + helpers = Counter(dims[g['ends'][r]]-dims[g['starts'][r]] + for r in range(w['public']) if r not in main) + paid.pop(0,None) + zero = helpers.pop(0,0) + return {'paid':dict(paid), 'helpers':dict(helpers), 'zero_helpers':zero, + 'owned_rank':sum(dims[a]-dims[b] for a,b,p,_ in g['moves'] if p)} diff --git a/research/p104-monotone-frame-ceiling/input/NOTICE-6398 b/research/p104-monotone-frame-ceiling/input/NOTICE-6398 new file mode 100644 index 0000000000..03e4ac5937 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/input/NOTICE-6398 @@ -0,0 +1,287 @@ +Mixed projector query root +Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0. No subagents. + +The new contribution is the normalized six-field root, its weighted binary +children, exact complete caller, full invoice and conditional bound. +The shortest central-involution query tuple is from the credited +five-call-projector-rigidity package by Ali Romman with OpenAI Codex. +The binary query extension is PR6348. The unchanged native source is PR6349. +The simple normalization identity is not claimed as a new general algebra theorem. +The native construction, scalar compression and inherited proof engines are +predecessor work. Their exact source notices follow. + +Source: research/balanced-query-tree/NOTICE + +Binary query-tree caller + +New construction, proof and code: Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0. No subagents. + +Cinna Davis supplies the productive path caller, query thinning and exact Laurent factorization. Ali Romman supplies the all-forward P104 binding in PR5544 and the retained-operand word in PR6167. The native word and interval/compiler lineage remain predecessor work. The new contribution is the four-field query extension, its tree composition, complete invoice and conditional price. + +The authenticated PR6167 notice follows. + +Retained-operand descent: Ali Romman, with substantial OpenAI Codex assistance. +Apache-2.0. + +The exact source word and price are from DreamingOfClouds, PR6144 and PR6143, +with Claude Opus 5.5 (Anthropic), via Claude Code. They extend PR6030 and PR5907. +The source reader is adapted from that work. The gzip portability controls +retain Douglas Colkitt and Maxime Fleury credit. + +The pinned PR5827 checker and all earlier source notices are retained below. + +Unread-register path resynthesis +Copyright 2026 Ali Romman. Apache-2.0. + +New construction, code, calculations and prose: Ali Romman with substantial +OpenAI Codex assistance. No subagents. All new checks are author-side. + +Mason Hale supplied PR5811, the complete two-block composition and the exact +role92 remapping and fourth accumulation. The earlier elementary accumulation, +retained-copy and second-block work is by Ali Romman. Joel Pulikkan supplied +R22. Dugongue supplied the concurrent baseline composition proof. Their source +notice and proof are retained byte for byte under inputs/pr5811-*. + +The source closure retains flyingviper, Cinna Davis, David Arias, Gareth +Davidson, Jacob Alan Sussman, zxv, Cao Yifan, Pierre-Loic Doulcet, thu-nmrc, +Scratch-net and all native/helper/controller/compiler/geometry/analytic +predecessors. The exact PR5786 source archive and its nested notices remain. +This package does not claim any predecessor construction as new. + +Exact PR5811 vendor/accumulated/NOTICE follows. Source head: +33d7760d0a6fea6394aefd014eea68f0af02a92e + +Accumulated native updates +New work: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +No subagents. The same assistant made the construction and the author checks. + +Concurrent scope: Discussion5125 now tests retained copies in the transposed +supplier. That route uses the same source/destination duality. This package +claims the three fixed forward-supplier groups and their complete result, +not the elementary accumulation identity or the broader transposed search. + +Exact source: PR5786 at 6b936acd68a7e0f9062037900f0fe9bc4b6fdd25. +The complete predecessor notice follows unchanged. + +Asynchronous helper hubs +New work: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. + +Retained native copies from PR5774 at 7d6ec0cdef31d0c37b7d2563e709979caa0d5eae. +All prior credit follows unchanged. + +Paid intermediate meetings and collective helper splits. +Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0. +No subagents. All checks are author-side calculations. + +The complete PR5718 notice follows unchanged. + +Four paid interior public-basis changes. +Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0. +No subagents. + +The following exact notice is retained from PR5660. + +New helper-subspace construction, proof and replay: +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +The same assistant performed the construction and author-side checks. + +Retained contributions: +flyingviper: PR5605 shared relay, PR5612 delayed relay, and the PR5608/PR5613 +relay/split compositions. +Scratch-net: PR5606 prepared-helper identities and target debit. +Pierre-Loic Doulcet (hexapode): PR5565 commuting-contact construction, used as the exact source. +Ali Romman: PR5602 fixed boundary relations, PR5601 three-contact frame +change, PR5580 contact step, +and PR5544/5545 complete P104 caller. +Joel Pulikkan (GamingPuzzled): PR5569 priced helper handoff. +DreamingOfClouds: PR5561 free helper endpoints and PR5568 search tools. +thu-nmrc: PR5551 contact improvements. +Gareth Davidson (bitplane): PR2305 source composition. +David Arias: PR2152 coalesced P104 word. +Cinna Davis: PR2219 private reuse and the controller lineage. +Jacob Alan Sussman: productive mixed-target recursion. +zxv and Cao Yifan: exact intervals, compiler and native/pricing lineage. +GamingPuzzled: PR5472 successor handoff, a predecessor to the split concept. +The retained source notices preserve earlier geometry, BIT and analytic credit. + +The elementary rank-one basis identity is not claimed as a new general +linear-algebra theorem. The contribution is its explicit independent-target +use on this P104 word and the complete conditional kappa improvement. + +New contribution: Ali Romman, with substantial OpenAI Codex assistance. +The four-gate source-version rewrite composes zxv PR5643 and Ali Romman PR5644. +The A2 method retains Chris R/Clear1nsight PR3998 and ToikY PR4019 credit. +The compensated triangle method retains Cinna Davis PR4692 credit. +No subagents were used for this contribution. Apache-2.0. + +Concurrent PR5650 by zxv publishes the PR5643/PR5644 composition. PR5651 by +Ali Romman corroborates it. This package credits that released composition +and contributes the further four-gate placement. + +New work: distributed scalar updates on an unobserved public field interval, the exact witness and complete conditional price. Prepared for Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0. The source archive retains all predecessor notices. + +Two-boundary main-assisted helper split and early cleanup: flyingviper with substantial +OpenAI Codex assistance, including independent AI review agents. Apache-2.0. + +Retained native-copy construction +Prepared by Ali Romman with substantial OpenAI Codex assistance. No subagents. +The paid-meeting source is PR5751 by Ali Romman with OpenAI Codex assistance. +The dimension-four span [265,194,164,75] is from Joel Pulikkan / GamingPuzzled +PR5739, with Claude Opus 5.5 assistance. The R22 block edits are not included. +The retained-copy mechanism, exact constructor, source-specific verification, +new price and proof are the contribution of this package. Apache-2.0. + +Exact PR5811 vendor/blocks/NOTICE follows. Source head: +33d7760d0a6fea6394aefd014eea68f0af02a92e + +The second block is PR5796 at4b30391e633ba0c7c674f03586ae97d443e9bc9c, by Ali Romman with substantial OpenAI Codex assistance, following Joel Pulikkan/GamingPuzzled R22. Exact notice, plan, proof and source words are retained in source/SECOND-* unchanged. + +The final immediate source is PR5786 asynchronous helper hubs by Ali Romman with substantial OpenAI Codex assistance. Its three temporal helper splits and complete completion parameters are retained and credited. The earlier PR5774-only composition is a superseded checkpoint, not a separate contribution. + +New complete retained-copy and R22 composition: Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0. + +The thirteen retained native conversions and changed helper targets are by Ali Romman with OpenAI Codex assistance in PR5774. The R22 block and rank-four basis are by Joel Pulikkan (GamingPuzzled), with Claude Opus5.5 assistance, in PR5739/5756. The complete source archive and R22 notice remain unchanged. + +The construction retains Ali Romman PR5751 paid intermediate meetings, flyingviper PR5718 boundary splits, Cinna Davis's productive controller, Jacob Alan Sussman's productive recursion, zxv and Cao Yifan's interval/compiler line, and every P104/native/BIT predecessor in the source notices. The source engines are reused under exact byte bindings, not claimed as new implementations. + +Source: research/p104-native-input-lifting/NOTICE + +Native helper input lifting +Copyright 2026 Ali Romman. Apache-2.0. + +New construction, code, calculations and prose: Ali Romman with substantial +OpenAI Codex assistance. No subagents. All new checks are author checks. + +The direct source is PR6314 at 36533d4d1b4e2dd8672427461b12de3001b85810, +by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. +It compresses the scalar word of PR6167 without a kappa change. +The four native input lifts and their complete conditional price are new here. +The basis identity is elementary linear algebra; no new general theorem +for that identity is claimed. + +The PR6167 source notice follows unchanged. Its nested source archive also +retains the earlier notices. + +Retained-operand descent: Ali Romman, with substantial OpenAI Codex assistance. +Apache-2.0. + +The exact source word and price are from DreamingOfClouds, PR6144 and PR6143, +with Claude Opus 5.5 (Anthropic), via Claude Code. They extend PR6030 and PR5907. +The source reader is adapted from that work. The gzip portability controls +retain Douglas Colkitt and Maxime Fleury credit. + +The pinned PR5827 checker and all earlier source notices are retained below. + +Unread-register path resynthesis +Copyright 2026 Ali Romman. Apache-2.0. + +New construction, code, calculations and prose: Ali Romman with substantial +OpenAI Codex assistance. No subagents. All new checks are author-side. + +Mason Hale supplied PR5811, the complete two-block composition and the exact +role92 remapping and fourth accumulation. The earlier elementary accumulation, +retained-copy and second-block work is by Ali Romman. Joel Pulikkan supplied +R22. Dugongue supplied the concurrent baseline composition proof. Their source +notice and proof are retained byte for byte under inputs/pr5811-*. + +The source closure retains flyingviper, Cinna Davis, David Arias, Gareth +Davidson, Jacob Alan Sussman, zxv, Cao Yifan, Pierre-Loic Doulcet, thu-nmrc, +Scratch-net and all native/helper/controller/compiler/geometry/analytic +predecessors. The exact PR5786 source archive and its nested notices remain. +This package does not claim any predecessor construction as new. + +Exact PR5811 vendor/accumulated/NOTICE follows. Source head: +33d7760d0a6fea6394aefd014eea68f0af02a92e + +Accumulated native updates +New work: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +No subagents. The same assistant made the construction and the author checks. + +Concurrent scope: Discussion5125 now tests retained copies in the transposed +supplier. That route uses the same source/destination duality. This package +claims the three fixed forward-supplier groups and their complete result, +not the elementary accumulation identity or the broader transposed search. + +Exact source: PR5786 at 6b936acd68a7e0f9062037900f0fe9bc4b6fdd25. +The complete predecessor notice follows unchanged. + +Asynchronous helper hubs +New work: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. + +Retained native copies from PR5774 at 7d6ec0cdef31d0c37b7d2563e709979caa0d5eae. +All prior credit follows unchanged. + +Paid intermediate meetings and collective helper splits. +Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0. +No subagents. All checks are author-side calculations. + +The complete PR5718 notice follows unchanged. + +Four paid interior public-basis changes. +Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0. +No subagents. + +The following exact notice is retained from PR5660. + +New helper-subspace construction, proof and replay: +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +The same assistant performed the construction and author-side checks. + +Retained contributions: +flyingviper: PR5605 shared relay, PR5612 delayed relay, and the PR5608/PR5613 +relay/split compositions. +Scratch-net: PR5606 prepared-helper identities and target debit. +Pierre-Loic Doulcet (hexapode): PR5565 commuting-contact construction, used as the exact source. +Ali Romman: PR5602 fixed boundary relations, PR5601 three-contact frame +change, PR5580 contact step, +and PR5544/5545 complete P104 caller. +Joel Pulikkan (GamingPuzzled): PR5569 priced helper handoff. +DreamingOfClouds: PR5561 free helper endpoints and PR5568 search tools. +thu-nmrc: PR5551 contact improvements. +Gareth Davidson (bitplane): PR2305 source composition. +David Arias: PR2152 coalesced P104 word. +Cinna Davis: PR2219 private reuse and the controller lineage. +Jacob Alan Sussman: productive mixed-target recursion. +zxv and Cao Yifan: exact intervals, compiler and native/pricing lineage. +GamingPuzzled: PR5472 successor handoff, a predecessor to the split concept. +The retained source notices preserve earlier geometry, BIT and analytic credit. + +The elementary rank-one basis identity is not claimed as a new general +linear-algebra theorem. The contribution is its explicit independent-target +use on this P104 word and the complete conditional kappa improvement. + +New contribution: Ali Romman, with substantial OpenAI Codex assistance. +The four-gate source-version rewrite composes zxv PR5643 and Ali Romman PR5644. +The A2 method retains Chris R/Clear1nsight PR3998 and ToikY PR4019 credit. +The compensated triangle method retains Cinna Davis PR4692 credit. +No subagents were used for this contribution. Apache-2.0. + +Concurrent PR5650 by zxv publishes the PR5643/PR5644 composition. PR5651 by +Ali Romman corroborates it. This package credits that released composition +and contributes the further four-gate placement. + +New work: distributed scalar updates on an unobserved public field interval, the exact witness and complete conditional price. Prepared for Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0. The source archive retains all predecessor notices. + +Two-boundary main-assisted helper split and early cleanup: flyingviper with substantial +OpenAI Codex assistance, including independent AI review agents. Apache-2.0. + +Retained native-copy construction +Prepared by Ali Romman with substantial OpenAI Codex assistance. No subagents. +The paid-meeting source is PR5751 by Ali Romman with OpenAI Codex assistance. +The dimension-four span [265,194,164,75] is from Joel Pulikkan / GamingPuzzled +PR5739, with Claude Opus 5.5 assistance. The R22 block edits are not included. +The retained-copy mechanism, exact constructor, source-specific verification, +new price and proof are the contribution of this package. Apache-2.0. + +Exact PR5811 vendor/blocks/NOTICE follows. Source head: +33d7760d0a6fea6394aefd014eea68f0af02a92e + +The second block is PR5796 at4b30391e633ba0c7c674f03586ae97d443e9bc9c, by Ali Romman with substantial OpenAI Codex assistance, following Joel Pulikkan/GamingPuzzled R22. Exact notice, plan, proof and source words are retained in source/SECOND-* unchanged. + +The final immediate source is PR5786 asynchronous helper hubs by Ali Romman with substantial OpenAI Codex assistance. Its three temporal helper splits and complete completion parameters are retained and credited. The earlier PR5774-only composition is a superseded checkpoint, not a separate contribution. + +New complete retained-copy and R22 composition: Ali Romman with substantial OpenAI Codex assistance. No subagents. Apache-2.0. + +The thirteen retained native conversions and changed helper targets are by Ali Romman with OpenAI Codex assistance in PR5774. The R22 block and rank-four basis are by Joel Pulikkan (GamingPuzzled), with Claude Opus5.5 assistance, in PR5739/5756. The complete source archive and R22 notice remain unchanged. + +The construction retains Ali Romman PR5751 paid intermediate meetings, flyingviper PR5718 boundary splits, Cinna Davis's productive controller, Jacob Alan Sussman's productive recursion, zxv and Cao Yifan's interval/compiler line, and every P104/native/BIT predecessor in the source notices. The source engines are reused under exact byte bindings, not claimed as new implementations. diff --git a/research/p104-monotone-frame-ceiling/input/NOTICE-6453 b/research/p104-monotone-frame-ceiling/input/NOTICE-6453 new file mode 100644 index 0000000000..28e0798625 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/input/NOTICE-6453 @@ -0,0 +1,18 @@ +New contribution: written by the Claude Sonnet 5.5 main session (Claude Code), with orchestrated subagent +assistance, for the user Erwan L. Apache-2.0. AI-assistance disclosure: the third-anchor construction (SPEC.json, +construct3.py), the parametrized merge copy, the exact zero-scratch check, the adapted verifier and this text were +produced by AI assistants under the user's direction; all new checks are author checks. + +Lineage (source and method, not authors of the new code): PR6417 (rohanarun) composes PR6399 and PR6398; PR6375 +(DreamingOfClouds); PR6372 and PR6353 (Ali Romman). merge_construct.py is a copy of the PR6399 constructor +(Rohan Arun with substantial OpenAI Codex assistance, Apache-2.0) with the cache label made a parameter. The PR6417 +notice follows byte for byte; the predecessor notices are retained byte-for-byte in the four source archives. + +New source-bound composition and portable package: Rohan Arun, with substantial OpenAI Codex assistance. Apache-2.0. + +Exact prior clean native merge: PR6399, Rohan Arun with substantial OpenAI Codex assistance. +Mixed six-field root, normalization, physical binding and price adapter: PR6398, Ali Romman with substantial OpenAI Codex assistance. +Shared-carrier input lifts: PR6375, DreamingOfClouds with Claude Opus5.5 via Claude Code. +All predecessor credit notices are retained byte-for-byte in native-source.tar and root-source.tar. +The P104, Gaussian/native, productive helper, BIT, compiler, precision, outer and original manuscript authors retain their attribution and source licenses. +No new claim is made for the mixed root itself or for classical rank-one inverse identities. diff --git a/research/p104-monotone-frame-ceiling/input/NOTICE-6529 b/research/p104-monotone-frame-ceiling/input/NOTICE-6529 new file mode 100644 index 0000000000..3d9b7e4c8b --- /dev/null +++ b/research/p104-monotone-frame-ceiling/input/NOTICE-6529 @@ -0,0 +1,21 @@ +Scaled transport fusion +Copyright 2026 Ali Romman. Apache-2.0. + +New construction and package: Ali Romman, with substantial OpenAI Codex +assistance. No subagents were used. + +The method extends Rohan Arun's PR6399 donor transport merge to intervals +with a scalar rescaling. His PR6417 supplies the clean mixed composition. +Erwan L., with the assistance disclosed in PR6453, supplies the third-anchor +stack used as the exact source. DreamingOfClouds supplies the PR6375 lifts +and retained native optimizations. Those contributions are not new here. + +Ali Romman's earlier contact basis and mixed caller retain their prior credit. +Cinna Davis, David Arias, Gareth Davidson, Jacob Alan Sussman, zxv, Cao Yifan, +Douglas Colkitt, and all retained native, helper, BIT, compiler and analytic +authors keep their credit. Tekalign's package exporter supplies source.tar. +The complete original notices and source attributions remain unchanged inside +that archive and its nested source archives. + +The exact source is CrocSwap/integer-mult-bounds PR6453, +b1a3e525793eca35c56324aa7ffd0b714cf606d4. diff --git a/research/p104-monotone-frame-ceiling/input/ROOT-PROOF.md b/research/p104-monotone-frame-ceiling/input/ROOT-PROOF.md new file mode 100644 index 0000000000..510bf9dba0 --- /dev/null +++ b/research/p104-monotone-frame-ceiling/input/ROOT-PROOF.md @@ -0,0 +1,222 @@ +# A mixed projector root for independent Gaussian outputs + +This construction changes the caller. It keeps the complete PR6349 native +source. Four binary query suppliers, with 7, 2, 2 and 7 native leaves, enter +a six-field root. The root has three thin queries and two individual +whole idle jobs. It gives six independent Gaussian outputs. The child +suppliers give another 28 independent main outputs. + +The conditional claim is kappa 0.00328467. The five inherited framework +assumptions remain. This is not an unconditional multiplication theorem, +a materialized uniform word or an independent scientific review. + +## 1. An ordinary normalization of the native query + +Write q = Q^-1 for the actual common central Gaussian. The native checker +verifies Q^2 = Q^-2 = X with absolute phase +1. X is an ordinary translation +and X^2 = I. These identities hold under the retained tensor implementation. +Define + +```text +lambda = (1+i)I/2 + (1-i)X/2, +lambda^-1 = (1-i)I/2 + (1+i)X/2, +S = lambda q. +``` + +Then lambda^2 = X, S^2 = I, and lambda^-1 S = q. All these operators commute +with the fresh stage products D. The normalization is applied to the complete +field, using its global translation X. Its scalar coefficients stay outside +tensor powers. In general S is not the tensor power of the one-column S. + +The translation has disjoint two-address orbits. On one orbit lambda has +matrix [[a,b],[b,a]], where a=(1+i)/2 and b=(1-i)/2. Four ordinary complex +scalar operations implement it in place: + +```text +x <- x - i*y +x <- a*x +y <- (1+i)*y +y <- y - i*x. +``` + +Conjugate each coefficient to implement the inverse. No owned field, paid +Gaussian job or productive input is added. The pair map is unitary; each +displayed operation has growth at most two in the maximum norm. There are +four operations per address pair. A loop over the existing translation +pairs has linear data work. The ordinary-work bill below includes these +operations and translations. + +A child supplies D U(q), where U(q)=I+q E01, on two arbitrary fields. +Conjugation by diag(lambda,1), followed by U(-1), gives +D U(S-1). A thin query has the same conversion and retains its paid H1 +charge. Thus the central-involution root below has an actual native +implementation. Its output S is converted back to q by lambda^-1. + +## 2. The six-field root + +Use the shortest-query normal form from +`research/five-call-projector-rigidity/PROOF.md`. For i=0,...,5 set u_i=e_i. +Set u_6=(-1,1,-1,1,-1,1), and (v_i)_j=sign(i-j). Let R_i=u_i v_i. +Each R_i is square-zero and has rank one. Direct exact multiplication gives + +```text +(I-2R_6) ... (I-2R_0) = -I_6. +``` + +`ROOT.json` specifies an invertible rational basis F, four stage projectors +P_i, and their native two-field bases C_i. Full sites are 0, 1, 5 and 6. +Sites 2, 3 and 4 are thin. In F coordinates the four projector masks are + +| Site | Output coordinates in the stage | Rank | Native leaves | Extra whole idles | +| --- | --- | ---: | ---: | --- | +| 0 | 0,1 | 2 | 7 | none | +| 1 | 0,1,5 | 3 | 2 | one H16 | +| 5 | 0,2,4 | 3 | 2 | one H16 | +| 6 | 0,2 | 2 | 7 | none | + +The exact checker verifies P_i^2=P_i, P_i R_i=R_i P_i=R_i and pairwise +commutation of the four P_i. It also verifies that P_i commutes with the +query prefix (I-2R_i)...(I-2R_0). These conditions permit each independent +stage factor to move past its own query prefix. The two central characters +then give the same diagonal stage product, with scalar factor +1 or -1. +The checker also multiplies every independent stage coefficient directly. + +At a full site, C_i takes R_i to E01 and P_i to diag(I_r,0). Apply the child +supplier to the first two coordinates. Apply D_i separately to each of the +remaining r-2 coordinates. These are exactly the charged whole idle jobs. +All basis changes are emitted as rational ADD, SCALE and SWAP operations. +At a thin site, the corresponding C_i converts one paid query to R_i. + +Apply F at entry and F^-1 at exit. Apply lambda^-1 to all six root outputs. +Their independent stage supports are + +```text +ABCD, AB, CD, empty, C, B. +``` + +For child sizes 7,2,2,7, the six Gaussian ranks are 145,73,73,1,17,17. +No root input is owned or initialized. The scalar basis maps are invertible. + +## 3. Binary children and full endpoint + +PR6348 supplies the unchanged four-field binary query extension. Each node +combines two query suppliers, one paid thin query and two whole idle jobs. +It returns another query supplier and two independent Gaussian outputs. +Its separate Laurent checker is replayed here under its source hash. + +The four child trees have 18 native leaves and 14 binary nodes. Their node +size counts are {2:8,3:2,4:2,7:2}. The root adds three thin queries. +The complete caller has 34 main fields per port, 18 distinct native stages, +17 thin queries, 14 paired whole idles and two individual whole idles. +Its 570 literal instructions include all basis maps and 20 normalizations. + +The full replay uses exact coefficients in Q(i)[Q,D_1,...,D_18]/(Q^4-1). +It checks all 1,156 input/output entries. Stage variables remain distinct. +It checks the complete Gaussian endpoint, not only the S=+1 and S=-1 cases. +Missing native calls, thin queries, paired idles, individual idles, +normalizations and central translations are rejected. + +Each native call receives 562 independent helper inputs. These are separate +from all main fields and from every other call's helpers. There are +18*562+34*104=13,652 productive fields. Calls execute in sequence. The 29 +owned labels, 32 native lifetimes and peak 13 fields remain as in PR6349. +Thin-query storage is reused between calls. Normalization adds no storage. + +The physical checker verifies all 7,488 native contacts, their absolute +phases, every output Gaussian and every paired or individual idle. The +fresh stage axes are orthonormal and fix the central port. The source word +receives its complete arbitrary-input, dense-transpose and native replay. + +## 4. Complete price and outer transfer + +The complete invoice has 66,988 paid jobs and 13,652 productive jobs. +Paid rank mass is 185,340; productive rank mass is 186,100. The surplus is +760. The maximum target rank is 145 and maximum paid rank is 56. The +emitted-event count agrees with a separate fixed-tree and root count. + +The strict raw crossing is in (0.00328468791927,0.00328468791928), between +adjacent points on the 10^-14 grid. `RESULT.json` records both rational +moment engines, completion, compiler and outer ledger. The claim is +0.00328467; the tested +saving is 0.003284675. The completion uses power 1024, cutoff 12, floor 13, +offset 1/7, reserve 10^-12 and decay 31/32. The logarithm power is 1100. +All local native, route and scalar costs remain charged. The new caller +uses a conservative 32 real scalar units per field per instruction, +including the normalization. Its complete ordinary bound is below 10^9. +The retained admitted BIT saving is .0034. All 54 strict and four functional +outer predicates must pass. + +The comparison is PR6353's published conditional .00315057. The present +source is PR6349; it does not consume PR6353's output-contracted word. +The source-relative gain is from the new caller, not new digits of an old +crossing. PR6353 and PR6350 retain their separate construction credit. + +The basis search checked 241,886,080 bounded bases modulo 65521. It found +695 support patterns. Floating searches selected the rank 2,3,3,2 pattern. +The exact rational checker then verified the chosen basis and every stage +coefficient. This finite search does not prove a global basis optimum. +The new child-size search selected 7,2,2,7. Section 5 gives an exact ceiling +for all binary children of this fixed root. + +## 5. A ceiling for every binary child size of this root + +This statement keeps the root and the PR6349 native source fixed. It permits +all positive integer child sizes and all binary tree shapes. Let H(r)=r^t, +t=1-kappa, and let L be the native paid-minus-helper moment per port. +Let C(n) be the minimum binary-node cost for n leaves. Then + +```text +C(1)=0, +C(n)=min_{1 <= a < n} (C(a)+C(n-a)+2H(8n)+1-2H(8n+1)). +``` + +For a pair with total n, the best root contribution is + +```text +G(n)=max_{1 <= b < n} + (H(8n+1)-n L-C(n-b)-C(b)+H(8b+1)-H(8b)). +``` + +For four positive children, with pair totals x and y, the complete margin +per port is H(8(x+y)+1)+G(x)+G(y)-2. The two pairs contain all root outputs, +three thin queries, both individual idles and every binary-node job. + +At kappa .00328469, `ceiling.py` uses rational outward intervals. It checks +all 4,851 binary splits below 100 leaves and all 4,656 root pair choices with +four through 99 leaves. Every margin upper bound is strictly negative. + +For larger totals f, concavity gives C(n) >= -(n-1) and +H(8b+1)-H(8b) <= H(9)-H(8). Jensen's inequality gives the upper bound + +```text +B(f)=H(8f+1)+2H(4f+1)-f(L-1)+2(H(9)-H(8))-6. +``` + +The checker proves B(100)<0 and B'(100)<0 with rational intervals. Since +t is strictly between zero and one, B' decreases. Thus B(f)<0 for every +real f>=100. No larger binary tree is omitted. + +The native deficit L has strictly positive derivative throughout +0<=kappa<=.0034. The checker proves this with an interval lower bound above +42. Each binary-node cost increases with kappa, while each positive H term +and each increment H(r+1)-H(r) decreases. Thus every root margin decreases +throughout the retained BIT admission range. Failure at .00328469 also +excludes every larger saving in that range. + +This fixed root cannot reach .00328469 with the retained native source. +The claim .00328467 is within .00000002 of that ceiling. This is a scoped +ceiling, not a bound for other roots, source words or query interfaces. + +## Scope and open next questions + +The inherited assumptions are native_all_columns, owned_erasure, compiler, +precision_tape and analytic_recovery. Finite operator and compiler checks +do not discharge their full quantified statements. The large uniform word +and rounded multiplier are not materialized. No global optimality or +practical speedup is claimed. The selected frontier stays unchanged. + +An independent reviewer should first check the global-field normalization, +the two charged rank-three projectors, and input independence. +Then replay the exact source and complete invoice. A later improvement can +change this root or combine it with a verified source, but must include +every changed productive debit, ordinary operation and compiler cost. diff --git a/research/p104-monotone-frame-ceiling/input/ROOT.json b/research/p104-monotone-frame-ceiling/input/ROOT.json new file mode 100644 index 0000000000..b59f1ea89a --- /dev/null +++ b/research/p104-monotone-frame-ceiling/input/ROOT.json @@ -0,0 +1,1319 @@ +{ + "status": "PASS_EXACT_MIXED_ROOT_AND_NORMALIZATION", + "fields": 6, + "full_sites": [ + 0, + 1, + 5, + 6 + ], + "thin_sites": [ + 2, + 3, + 4 + ], + "input_basis": [ + [ + "2", + "-1", + "1", + "0", + "0", + "0" + ], + [ + "-2", + "2", + "-1", + "0", + "0", + "1" + ], + [ + "1", + "-1", + "0", + "1", + "1", + "-1" + ], + [ + "-2", + "2", + "-1", + "0", + "0", + "2" + ], + [ + "2", + "-2", + "1", + "-1", + "0", + "-2" + ], + [ + "0", + "0", + "1", + "0", + "-1", + "0" + ] + ], + "output_basis": [ + [ + "1", + "1", + "-1", + "-2", + "-1", + "-1" + ], + [ + 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zRM*a8&|O_a8rijpLW3j}N#&hR&LS&$R$1>0*}Z=DZAI;2TVu!5p?u8W?*2IK{*~z4 z?68j9f3v^10^Wix1HeHE#Sn^mS(F{tvQuoRpShFb;gc$ro};FOW+;s@JPM62Laj<=P2ZSG z*;-$!POWNMHKZyAtxESl2Htwsw500%9;^>BS|4J#iqZNI!&QyehZ?SGv_8~ueW>xO zhN~K_4>epLYP_o9`cUKbp$050j%#lH=kfCK)6ehTzP^0p$KqdK|NYDRH!t73e2k$y x-n_nkdjIL&o8Nx<*W>4RA720Sc>VDHs5Om2LPYV|IYvb literal 0 HcmV?d00001 diff --git a/research/p104-monotone-frame-ceiling/model.py b/research/p104-monotone-frame-ceiling/model.py new file mode 100644 index 0000000000..dad6c8ba6a --- /dev/null +++ b/research/p104-monotone-frame-ceiling/model.py @@ -0,0 +1,72 @@ +"""Build exact local constraints for the frame tariff. + +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +""" +from collections import Counter +from itertools import combinations + +from frames import bounds, dimension + + +def model(w,g,k): + lo,hi=bounds(w,g);n=len(lo) + labels=[list(range(dimension(a),dimension(b)+1)) for a,b in zip(lo,hi)] + variables=[];cost=[];nodes=[];eq=[];ub=[];tariffs=[] + for i,values in enumerate(labels): + table={} + for d in values: + table[d]=len(cost);variables.append(('node',i,d));cost.append(0.) + nodes.append(table);eq.append((dict.fromkeys(table.values(),1),1)) + weights=Counter();successors=[set() for _ in range(n)] + for a,b,p,_ in g['moves']: + if p: + a,b=b,a + weights[a,b]+=1 + if a!=b: + successors[a].add(b) + main=set(w['live'])|{r+112 for r in w['live']} + for r in range(w['public']): + if r not in main: + weights[g['starts'][r],g['ends'][r]]-=1 + for (a,b),weight in sorted(weights.items()): + if a==b: + continue + floor=dimension(lo[b])-dimension(lo[b]&hi[a]) + table={} + for x in labels[a]: + for y in labels[b]: + if y-x>=floor: + rank=y-x;index=len(cost);table[x,y]=index + variables.append(('edge',a,b,x,y)) + cost.append(0 if k is None else weight*rank**(1-k)) + tariffs.append((index,weight,rank)) + for x in labels[a]: + row={i:1 for (xx,y),i in table.items() if xx==x} + row[nodes[a][x]]=-1;eq.append((row,0)) + for y in labels[b]: + row={i:1 for (x,yy),i in table.items() if yy==y} + row[nodes[b][y]]=-1;eq.append((row,0)) + + def dimensions(coeff,rhs): + row=Counter() + for a,c in coeff.items(): + for d,i in nodes[a].items(): + row[i]+=c*d + ub.append(({i:x for i,x in row.items() if x},rhs)) + + reachable=[set() for _ in range(n)];predecessors=[set() for _ in range(n)] + for a in sorted(range(n),key=lambda i:-len(w['frames'][g['old'][i]])): + for b in successors[a]: + reachable[a].add(b);reachable[a].update(reachable[b]);predecessors[b].add(a) + for a,targets in enumerate(reachable): + for b in sorted(targets): + floor=dimension(lo[b])-dimension(lo[b]&hi[a]) + if floor>0 and (a,b) not in weights: + dimensions({a:1,b:-1},-floor) + for c in range(n): + for a,b in combinations(sorted(predecessors[c]),2): + dimensions({a:1,b:1,c:-1},dimension(hi[a]&hi[b])) + for a,b in combinations(sorted(successors[c]),2): + floor=dimension(lo[a])+dimension(lo[b])-dimension(lo[a]&lo[b]) + dimensions({a:-1,b:-1,c:1},-floor) + return variables,cost,eq,ub,tariffs diff --git a/research/p104-monotone-frame-ceiling/verify.py b/research/p104-monotone-frame-ceiling/verify.py new file mode 100644 index 0000000000..a25bf96c3f --- /dev/null +++ b/research/p104-monotone-frame-ceiling/verify.py @@ -0,0 +1,231 @@ +"""Check the rational frame-cost certificate and the all-size caller bound. + +No solver or floating arithmetic is used to decide a proof condition. +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +import argparse +import gzip +import json +from pathlib import Path +import sys + +sys.path.insert(0, str(Path(__file__).resolve().parent)) + +from exact import UNIT, log_rank, power +from model import model +from frames import bounds, dimension, graph, invoice, span + +HERE = Path(__file__).resolve().parent +HELPERS = {1:14, 2:8, 3:4, 4:107, 5:15, 6:3, 7:9, 8:39, 9:363} + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def model_hash(variables, eq, ub, tariffs): + payload = [variables, [[sorted(a.items()), b] for a, b in eq], + [[sorted(a.items()), b] for a, b in ub], tariffs] + return sha256(json.dumps(payload, separators=(',', ':')).encode()).hexdigest() + + +def dual_bound(variables, eq, ub, tariffs, certificate, kappa, derivative=False): + need(certificate['model_sha256'] == model_hash(variables, eq, ub, tariffs), + 'The dual must bind the reconstructed model') + scale = certificate['scale'] + need(isinstance(scale, int) and scale > 0, 'A positive integer dual scale') + residual = [0]*len(variables) + for j, weight, rank in tariffs: + if derivative: + if not rank: + value = 0 + elif weight >= 0: + value = -weight*rank*log_rank(rank)[1] + else: + value = (-weight)*power(rank, Q('.0034'))[0]*log_rank(rank)[0]//UNIT + else: + lo, hi = power(rank, kappa) + value = weight*(lo if weight >= 0 else hi) + residual[j] = scale*value + rhs = 0 + for name, rows in [('eq', eq), ('ub', ub)]: + seen = set() + for i, value in certificate[name]: + need(isinstance(i, int) and 0 <= i < len(rows) and i not in seen, + 'Each dual row index must be valid and distinct') + need(isinstance(value, int), 'Each dual coefficient must be an integer') + need(name == 'eq' or value <= 0, 'Upper-bound dual signs must be nonpositive') + seen.add(i) + row, b = rows[i] + rhs += UNIT*value*b + for j, a in row.items(): + residual[j] -= UNIT*value*a + correction = sum(min(0, r) for r in residual) + return Q(rhs+correction, UNIT*scale), Q(correction, UNIT*scale) + + +def source_model_control(variables, eq, ub, tariffs, dims, source): + x = [] + for v in variables: + if v[0] == 'node': + x.append(int(dims[v[1]] == v[2])) + else: + x.append(int((dims[v[1]], dims[v[2]]) == (v[3], v[4]))) + need(all(sum(a*x[j] for j, a in row.items()) == rhs for row, rhs in eq), + 'The original word must satisfy every equality') + need(all(sum(a*x[j] for j, a in row.items()) <= rhs for row, rhs in ub), + 'The original word must satisfy every inequality') + histogram = Counter() + for j, w, r in tariffs: + histogram[r] += w*x[j] + expected = Counter(source['paid']) + expected.subtract(source['helpers']) + histogram.pop(0, None) + need(dict(+histogram) == dict(+expected) and dict(-histogram) == dict(-expected), + 'The model must reproduce the complete original signed invoice') + + +def caller_bound(native_lower, kappa): + # The recurrence and tail are from PR6398, section 5. All helper + # targets, binary queries and paired or individual idles remain paid. + tail = 100 + h = {r: tuple(Q(x, UNIT) for x in power(r, kappa)) for r in range(1, 8*tail+2)} + lower = [Q(0)]*tail + binary_splits = 0 + for n in range(2, tail): + node = 2*h[8*n][0]+1-2*h[8*n+1][1] + lower[n] = min(lower[a]+lower[n-a]+node for a in range(1, n)) + binary_splits += n-1 + pairs = {} + for n in range(2, tail-1): + pairs[n] = max(h[8*n+1][1]-n*native_lower/104-lower[n-b]-lower[b] + +h[8*b+1][1]-h[8*b][0] for b in range(1, n)) + margins = [(h[8*n+1][1]+pairs[a]+pairs[n-a]-2, n, a) + for n in range(4, tail) for a in range(2, n-1)] + finite, total, first_pair = max(margins) + bound = (h[8*tail+1][1]+2*h[4*tail+1][1]-tail*(native_lower/104-1) + +2*(h[9][1]-h[8][0])-6) + derivative = (8*(1-kappa)*(h[8*tail+1][1]/(8*tail+1) + +h[4*tail+1][1]/(4*tail+1))-(native_lower/104-1)) + need(finite < 0, 'Every finite root must have negative raw margin') + need(bound < 0 and derivative < 0, 'The concave tail must be negative and decreasing') + return {'binary_splits': binary_splits, 'root_pair_checks': len(margins), + 'finite_margin_upper': str(finite), 'finite_total': total, + 'finite_first_pair': first_pair, 'tail_from': tail, + 'tail_margin_upper': str(bound), 'tail_derivative_upper': str(derivative)} + + +def positive_caller_control(source, kappa): + """Check the retained 7,2,2,7 caller with outward lower margins.""" + h = {r: tuple(Q(x, UNIT) for x in power(r, kappa)) for r in range(1, 146)} + native_upper = (sum(c*h[r][1] for r, c in source['paid'].items()) + -sum(c*h[r][0] for r, c in source['helpers'].items()))/104 + upper = [Q(0)]*8 + for n in range(2, 8): + node = 2*h[8*n][1]+1-2*h[8*n+1][0] + upper[n] = min(upper[a]+upper[n-a]+node for a in range(1, n)) + margin = (h[145][0]+2*h[73][0]+1+2*h[17][0] + -18*native_upper-2*upper[7]-2*upper[2]-3-2*h[16][1]) + need(margin > 0, 'The source must pass its published raw caller price') + return str(margin) + + +def reject(action, message): + try: + action() + except ValueError: + return + raise ValueError('A negative control passed: '+message) + + +def controls(variables, eq, ub, tariffs, certificate, dims, source, kappa): + bad = dict(certificate, model_sha256='0'*64) + reject(lambda: dual_bound(variables, eq, ub, tariffs, bad, kappa), 'wrong model hash') + bad = dict(certificate, ub=[[0, 1]]) + reject(lambda: dual_bound(variables, eq, ub, tariffs, bad, kappa), 'wrong dual sign') + bad = dict(certificate, eq=[[0, 0], [0, 0]]) + reject(lambda: dual_bound(variables, eq, ub, tariffs, bad, kappa), 'duplicate row') + bad = dict(certificate, eq=[[len(eq), 0]]) + reject(lambda: dual_bound(variables, eq, ub, tariffs, bad, kappa), 'invalid row') + changed = dict(source, paid=dict(source['paid'])) + changed['paid'][1] -= 1 + reject(lambda: source_model_control(variables, eq, ub, tariffs, dims, changed), + 'one omitted paid rank-one call') + changed = dict(source, helpers=dict(source['helpers'])) + changed['helpers'][9] += 1 + reject(lambda: source_model_control(variables, eq, ub, tariffs, dims, changed), + 'one unpaid productive target') + reject(lambda: caller_bound(Q(0), kappa), 'zero native bill') + return 7 + + +def source_bytes(path, expected): + raw = path.read_bytes() + need(sha256(raw).hexdigest() == expected, 'The exact source word hash') + return raw + + +def run_case(spec): + raw = source_bytes(HERE/'input'/spec['word'], spec['sha256']) + w = json.loads(gzip.decompress(raw)) + g = graph(w) + lo, hi = bounds(w, g) + old = [span(w['frames'][f]) for f in g['old']] + source = invoice(w, g, old) + need(source['owned_rank'] == 56, 'The source owned rank control') + need(source['paid'] == spec['paid'], 'The source paid invoice control') + need(source['helpers'] == HELPERS, 'The source helper invoice control') + variables, _, eq, ub, tariffs = model(w, g, None) + dims = list(map(dimension, old)) + source_model_control(variables, eq, ub, tariffs, dims, source) + kappa = Q(spec['kappa']) + certificate = json.loads((HERE/spec['dual']).read_text()) + lower, correction = dual_bound(variables, eq, ub, tariffs, certificate, kappa) + derivative_certificate = json.loads((HERE/spec['derivative']).read_text()) + derivative_lower, _ = dual_bound(variables, eq, ub, tariffs, derivative_certificate, kappa, True) + need(derivative_lower > 3866, 'Every legal native deficit must increase up to the BIT cap') + source_upper = Q(sum(c*power(r, kappa)[1] for r, c in source['paid'].items()) + -sum(c*power(r, kappa)[0] for r, c in source['helpers'].items()), UNIT) + need(source_upper-lower < Q('.055'), 'The maximum native saving bound') + return {'source': spec['name'], 'kappa': str(kappa), 'source_sha256': spec['sha256'], + 'components': len(g['old']), 'flexible_components': sum(a != b for a, b in zip(lo, hi)), + 'variables': len(variables), 'equalities': len(eq), 'inequalities': len(ub), + 'model_sha256': certificate['model_sha256'], 'native_cost_lower': str(lower), + 'native_saving_upper': str(source_upper-lower), 'dual_correction': str(correction), + 'native_derivative_lower': str(derivative_lower), 'saving_cap': '17/5000', + 'positive_caller_margin_lower': positive_caller_control(source, Q(spec['source_kappa'])), + 'negative_controls': controls(variables, eq, ub, tariffs, certificate, dims, source, kappa), + 'caller': caller_bound(lower, kappa)} + + +def run(): + sources = json.loads((HERE/'SOURCES.json').read_text()) + for row in sources['retained_predecessors']: + source_bytes(HERE/row['retained_path'], row['sha256']) + specs = json.loads((HERE/'CASES.json').read_text()) + for spec in specs: + spec['paid'] = {int(r): c for r, c in spec['paid'].items()} + results = [run_case(spec) for spec in specs] + need(Q(results[0]['kappa']) <= Q(results[1]['kappa']), 'The source ceiling order') + need(Q(results[0]['native_cost_lower']) > Q(results[1]['native_cost_lower']), + 'The latest bound must also cover the earlier source by monotonicity') + return {'status': 'PASS_EXACT_MONOTONE_FRAME_CALLER_EXCLUSION', + 'mixed_source_ceiling': results[1]['kappa'], 'cases': results} + + + +if __name__ == '__main__': + parser = argparse.ArgumentParser() + parser.add_argument('--write', action='store_true') + args = parser.parse_args() + result = run() + text = json.dumps(result, indent=2)+'\n' + if args.write: + (HERE/'RESULT.json').write_text(text) + else: + need(result == json.loads((HERE/'RESULT.json').read_text()), 'The exact retained receipt') + print(text, end='') From 6a3e041f250fd777f9b27c208853fb911e7a8d11 Mon Sep 17 00:00:00 2001 From: Ali Romman <226877337+aliromman@users.noreply.github.com> Date: Mon, 2 Nov 2026 22:13:22 -0600 Subject: [PATCH 2/5] Record sealed replay of both P104 frame ceilings --- research/p104-monotone-frame-ceiling/VALIDATION.md | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/research/p104-monotone-frame-ceiling/VALIDATION.md b/research/p104-monotone-frame-ceiling/VALIDATION.md index f0a5fb249b..1a2608f37b 100644 --- a/research/p104-monotone-frame-ceiling/VALIDATION.md +++ b/research/p104-monotone-frame-ceiling/VALIDATION.md @@ -23,8 +23,13 @@ The source models and four duals were discovered with NumPy 2.5.4 and SciPy by the exact verifier. `discover.py` is an optional numerical helper; its output remains provisional until the exact verifier checks it. -Metadata and exact source pins passed. Publication-byte, official-export and -sealed replay results will be added after the scientific files are committed. +Metadata, exact source pins and all 25 committed file bytes and modes passed +at `6053794e49861d7c46bb4aee8ef34606633658c9`. The official export and its +integrity check passed. Both CPython 3.12.13 replays passed in a read-only, +Git-free export with an empty PATH. The normal and optimized outputs were +identical. All 26 export files retained their bytes and modes. The stdout +SHA256 was `f47fdb60a7fb046e9aedabfe421da9f169f4cbaa331cc26360723cb3c8f35179`. +Only this validation prose changed after that export. The checkout is shallow. Full-history ancestry is not claimed. Every pending predecessor byte required by the proof is retained and checked directly. From f94fc3f2e2c53bdd629a38d0a4b8a962829d43ea Mon Sep 17 00:00:00 2001 From: Ali Romman <226877337+aliromman@users.noreply.github.com> Date: Mon, 2 Nov 2026 22:32:40 -0600 Subject: [PATCH 3/5] Bound every fresh-stage root child by 22 leaves and 21 at the one-percent target --- research/p104-monotone-frame-ceiling/PROOF.md | 71 ++++++++++++++++++- .../p104-monotone-frame-ceiling/README.md | 8 +++ .../p104-monotone-frame-ceiling/RESULT.json | 18 ++++- .../p104-monotone-frame-ceiling/TASK.json | 4 +- .../p104-monotone-frame-ceiling/VALIDATION.md | 7 +- .../p104-monotone-frame-ceiling/VERIFY.json | 2 +- .../p104-monotone-frame-ceiling/verify.py | 23 +++++- 7 files changed, 125 insertions(+), 8 deletions(-) diff --git a/research/p104-monotone-frame-ceiling/PROOF.md b/research/p104-monotone-frame-ceiling/PROOF.md index 5f8f05eb56..1ea231c077 100644 --- a/research/p104-monotone-frame-ceiling/PROOF.md +++ b/research/p104-monotone-frame-ceiling/PROOF.md @@ -221,7 +221,76 @@ decreases through kappa=.0034. The exclusions extend from their tested ceilings to that cap, even if one chooses a different tree or frame assignment at each kappa. -## 6. Interpretation and limits +## 6. A child-size cap for any fresh-stage root + +The next bound does not fix the PR6398 root. Fix any root with m full sites. +Site i receives a binary child of size n_i. Its stage occurs on r_i output +fields, where r_i>=2. Each occurrence contributes rank 8*n_i. Each output is +a product of distinct fresh stages and has a fixed additional rank d_j>=0. +The root pays exactly r_i-2 additional whole idles H(8*n_i) for that site. +Its other query charges do not depend on the child sizes. This includes a +root with paid overwrite queries and a root with ordinary thin shears. + +Use the uniform lower bound L on the native paid-minus-helper cost per port. +Its favorable raw margin has the form + +```text +F(n)=sum_j H(d_j+8 sum_(i in M_j) n_i) + -sum_i [(r_i-2)H(8*n_i)+n_i L+C(n_i)]-fixed_query_cost. +``` + +There are r_i sets M_j that contain i. For any such output its current rank +is at least 8*n_i. By concavity, its increase when n_i increases by one is +at most Delta_i=H(8*(n_i+1))-H(8*n_i). The r_i output increases, minus the +r_i-2 idle increases, are therefore at most 2*Delta_i. This cancellation +is why the bound does not depend on root size, masks or projector ranks. + +For every n>=1, `C(n+1)-C(n)>=-1`. To prove this, take a minimum-cost binary +tree on n+1 leaves and delete one leaf. Delete its parent and attach its +sibling in the parent's place. The removed parent cost is at least -1. +Every remaining ancestor loses one leaf, so its cost can only decrease: +`2H(8v)+1-2H(8v+1)` increases with v by concavity. The pruned tree is a legal +n-leaf tree and has cost at least C(n). This proves the claim for every n, +without a bound on tree size. + +It follows that, for any fixed values of the other children, + +```text +F(...,n_i+1,...)-F(...,n_i,...) + <= 2[H(8*(n_i+1))-H(8*n_i)]-L+1. +``` + +The right-hand side decreases with n_i. At kappa=.003293, the exact native +lower bound from Section 4 makes it less than -0.0004245 at n_i=22. +Thus any child larger than 22 can be reduced to 22 with a strict increase +in the favorable raw margin. This can be done one child at a time. + +The sign is unchanged for a coupled site with a_i main pairs. Such a site +has native and child charge `a_i*(n_i L+C(n_i))` and idle charge +`(r_i-2*a_i)H(8*n_i)`. Its increment bound is a_i times the displayed bound. +Thus the same caps apply at every coupled width with this complete bill. + +At the one-percent comparison target +`.003288299*101/100=.00332118199`, use the native lower bound at .003293 +plus the smaller of the two uniform derivative bounds times the increment +in kappa. The same expression is less than -0.0018422 at n_i=21. Thus 21 +is a sufficient child cap at this target. No minimality is claimed for +either cap. Since H(r+8)-H(r) decreases with kappa for r>=8, and the native +deficit increases, each cap remains valid up to the retained .0034 cap. + +The result concerns a favorable raw-budget relaxation. The lower native +bound need not be attained by an actual variant. A positive complete caller +would have a positive raw margin, hence a positive value in this relaxation. +If every capped case has nonpositive relaxed margin, no larger child can +repair it. The statement does not compare the ordinary or compiler costs +of two positive constructions. It does not cover repeated or noncommuting +stage factors, partial-stage idles or a different binary child contract. + +The source words, helpers, normalization and binary module remain credited +to their original authors. The general per-child cancellation and pruning +bound are the additional cost consequence proved here. + +## 7. Interpretation and limits The latest source comparison is PR6529's reported conditional .003288299. The ratio `.003293/.003288299-1` is less than .00143. Thus the whole stated diff --git a/research/p104-monotone-frame-ceiling/README.md b/research/p104-monotone-frame-ceiling/README.md index 04a5678b30..bac6528c4f 100644 --- a/research/p104-monotone-frame-ceiling/README.md +++ b/research/p104-monotone-frame-ceiling/README.md @@ -9,6 +9,14 @@ allowed native variant at each leaf. Thus this class has less than **0.143 percent** headroom over PR6529's reported 0.003288299. At the 0.003293 ceiling, different leaves can also use different source words. +A second result applies to **any root in the stated fresh-stage billing +model**, not only this fixed root. Its favorable raw-budget search needs at +most **22 leaves per binary child** for kappa>=0.003293, or **21 leaves** at +the current one-percent target 0.00332118199. The proof gives a bound on each +child increment, with no bound on the number of root fields or full sites. +This converts an unbounded child-size search into a finite one. It does not +provide a new root or a positive invoice. See `PROOF.md`, Section 6. + The proof keeps the scalar events, contact equalities, public forward moves, owned reverse moves and main boundaries. It permits all binary subspaces at all contacts, including new subspaces absent from the source. Helper input diff --git a/research/p104-monotone-frame-ceiling/RESULT.json b/research/p104-monotone-frame-ceiling/RESULT.json index bb0d56e457..d1336d3b3a 100644 --- a/research/p104-monotone-frame-ceiling/RESULT.json +++ b/research/p104-monotone-frame-ceiling/RESULT.json @@ -58,5 +58,21 @@ "tail_derivative_upper": "-1891746448077179841145363106619264370736425081803518810672287/31784714772709393369735640809338211919555038820696064000000000" } } - ] + ], + "general_child_caps": { + "scope": "Any root with fresh stage incidence and charged whole idles; raw budget only", + "tiers": [ + { + "kappa_from": "3293/1000000", + "child_cap": 22, + "increment_upper": "-126030930353210217593602209592244819027390857840481911/296867520082839655260123481645494988367611297792000000000" + }, + { + "kappa_from": "332118199/100000000000", + "child_cap": 21, + "increment_upper": "-21035220080060503658075860064087401742810151737762180986911084027/11417981541647679048466287755595961091061972992000000000000000000000" + } + ], + "extra_negative_controls": 1 + } } diff --git a/research/p104-monotone-frame-ceiling/TASK.json b/research/p104-monotone-frame-ceiling/TASK.json index 87f7bebad7..4a6f5a8547 100644 --- a/research/p104-monotone-frame-ceiling/TASK.json +++ b/research/p104-monotone-frame-ceiling/TASK.json @@ -97,13 +97,13 @@ ] }, "scope_status": "changed", - "actual_scope": "Prove an exact all-size caller ceiling for all simultaneous monotone contact subspaces on two fixed P104 scalar words, with variable helper boundaries and different variants at different native leaves.", + "actual_scope": "Prove an exact all-size caller ceiling for all simultaneous monotone contact subspaces on two fixed P104 scalar words, with variable helper boundaries and different variants at different native leaves. Prove a universal per-child size cap for any root with the stated fresh-stage and whole-idle invoice, including coupled sites.", "scope_change_reason": "The user requires a large complete kappa gain. Bounded frame searches did not give one. This proof decides whether the entire changed-frame contract can reach that gain before another construction search. The current PR6529 source is included after the live refresh.", "bottleneck_link": "The signed local invoice includes every paid MOVE and changed productive helper. The fixed mixed root and every binary child size are then priced. This excludes frame-only optimization as a route to a large gain in the stated model.", "budget_minutes": 360, "stop_rule": "Retain the exact scoped certificate and its controls. Do not call the negative result a kappa improvement or completion of the donor objective.", "outcome": "negative-result", - "evidence": "PROOF.md, RESULT.json, four exact dual certificates, two source words and verify.py.", + "evidence": "PROOF.md, RESULT.json, four exact dual certificates, two source words, the general child-cap inequalities and verify.py.", "author": "Ali Romman with substantial OpenAI Codex assistance", "ai_assistance": "Substantial OpenAI Codex assistance for research, code, proof, verification and prose. No subagents or independent human review.", "open_questions": [ diff --git a/research/p104-monotone-frame-ceiling/VALIDATION.md b/research/p104-monotone-frame-ceiling/VALIDATION.md index 1a2608f37b..af3609083f 100644 --- a/research/p104-monotone-frame-ceiling/VALIDATION.md +++ b/research/p104-monotone-frame-ceiling/VALIDATION.md @@ -10,7 +10,7 @@ session; the selected cold replay was not run. The exact direct normal and optimized replays passed under CPython 3.15.2. The registered normal and optimized replays passed under CPython 3.12.13 on macOS arm64, using only the standard library. They check both source systems, -the original signed invoices, fourteen rejection controls, both positive +the original signed invoices, fifteen rejection controls, both positive source callers, every finite caller case and both analytic tails. Five separate optimized-process corruptions were rejected: one byte in each @@ -29,7 +29,10 @@ integrity check passed. Both CPython 3.12.13 replays passed in a read-only, Git-free export with an empty PATH. The normal and optimized outputs were identical. All 26 export files retained their bytes and modes. The stdout SHA256 was `f47fdb60a7fb046e9aedabfe421da9f169f4cbaa331cc26360723cb3c8f35179`. -Only this validation prose changed after that export. +That export covered the original frame and fixed-root certificate. +The later Section 6 addendum gives the universal 22/21-leaf caps, including +coupled sites. Its updated receipt and both registered modes passed under +CPython 3.12.13. A new committed export will be checked for that addendum. The checkout is shallow. Full-history ancestry is not claimed. Every pending predecessor byte required by the proof is retained and checked directly. diff --git a/research/p104-monotone-frame-ceiling/VERIFY.json b/research/p104-monotone-frame-ceiling/VERIFY.json index 3d1a880cf7..8ff8627543 100644 --- a/research/p104-monotone-frame-ceiling/VERIFY.json +++ b/research/p104-monotone-frame-ceiling/VERIFY.json @@ -13,7 +13,7 @@ "{package}/verify.py" ], "timeout_seconds": 240, - "checks": "Rebuild both exact models; check source bytes, all original constraints and complete signed invoices; verify rational dual bounds, derivative bounds, positive callers and fourteen rejection controls; check all finite root sizes and both all-size tails." + "checks": "Rebuild both exact models; check source bytes, all original constraints and complete signed invoices; verify rational dual and derivative bounds, positive callers and fifteen rejection controls; check all finite fixed-root sizes, both all-size tails and the general 22/21-leaf child caps." }, { "argv": [ diff --git a/research/p104-monotone-frame-ceiling/verify.py b/research/p104-monotone-frame-ceiling/verify.py index a25bf96c3f..901403e7cb 100644 --- a/research/p104-monotone-frame-ceiling/verify.py +++ b/research/p104-monotone-frame-ceiling/verify.py @@ -169,6 +169,26 @@ def source_bytes(path, expected): return raw +def child_cap(native_per_port_lower, kappa, cap): + upper = (2*Q(power(8*(cap+1), kappa)[1]-power(8*cap, kappa)[0], UNIT) + -native_per_port_lower+1) + need(upper < 0, 'The universal child increment must be negative') + return {'kappa_from': str(kappa), 'child_cap': cap, 'increment_upper': str(upper)} + + +def general_child_caps(results): + base = Q(results[1]['native_cost_lower']) + k0 = Q(results[1]['kappa']) + derivative = min(Q(row['native_derivative_lower']) for row in results) + rows = [] + for kappa, cap in [(k0, 22), (Q('.003288299')*Q(101, 100), 21)]: + native_lower = (base+derivative*(kappa-k0))/104 + rows.append(child_cap(native_lower, kappa, cap)) + reject(lambda: child_cap(Q(0), k0, 22), 'a free native word in the child cap') + return {'scope': 'Any root with fresh stage incidence and charged whole idles; raw budget only', + 'tiers': rows, 'extra_negative_controls': 1} + + def run_case(spec): raw = source_bytes(HERE/'input'/spec['word'], spec['sha256']) w = json.loads(gzip.decompress(raw)) @@ -214,7 +234,8 @@ def run(): need(Q(results[0]['native_cost_lower']) > Q(results[1]['native_cost_lower']), 'The latest bound must also cover the earlier source by monotonicity') return {'status': 'PASS_EXACT_MONOTONE_FRAME_CALLER_EXCLUSION', - 'mixed_source_ceiling': results[1]['kappa'], 'cases': results} + 'mixed_source_ceiling': results[1]['kappa'], 'cases': results, + 'general_child_caps': general_child_caps(results)} From 72d8916c22463fac0d02f1c992a480ce83ec1791 Mon Sep 17 00:00:00 2001 From: Ali Romman <226877337+aliromman@users.noreply.github.com> Date: Mon, 2 Nov 2026 22:33:31 -0600 Subject: [PATCH 4/5] Record sealed verification of the general child-size bound --- research/p104-monotone-frame-ceiling/README.md | 2 +- research/p104-monotone-frame-ceiling/VALIDATION.md | 7 ++++++- 2 files changed, 7 insertions(+), 2 deletions(-) diff --git a/research/p104-monotone-frame-ceiling/README.md b/research/p104-monotone-frame-ceiling/README.md index bac6528c4f..0a80abaf2e 100644 --- a/research/p104-monotone-frame-ceiling/README.md +++ b/research/p104-monotone-frame-ceiling/README.md @@ -11,7 +11,7 @@ At the 0.003293 ceiling, different leaves can also use different source words. A second result applies to **any root in the stated fresh-stage billing model**, not only this fixed root. Its favorable raw-budget search needs at -most **22 leaves per binary child** for kappa>=0.003293, or **21 leaves** at +most **22 leaves per binary child** for 0.003293<=kappa<=0.0034, or **21 leaves** at the current one-percent target 0.00332118199. The proof gives a bound on each child increment, with no bound on the number of root fields or full sites. This converts an unbounded child-size search into a finite one. It does not diff --git a/research/p104-monotone-frame-ceiling/VALIDATION.md b/research/p104-monotone-frame-ceiling/VALIDATION.md index af3609083f..81d83ca39c 100644 --- a/research/p104-monotone-frame-ceiling/VALIDATION.md +++ b/research/p104-monotone-frame-ceiling/VALIDATION.md @@ -32,7 +32,12 @@ SHA256 was `f47fdb60a7fb046e9aedabfe421da9f169f4cbaa331cc26360723cb3c8f35179`. That export covered the original frame and fixed-root certificate. The later Section 6 addendum gives the universal 22/21-leaf caps, including coupled sites. Its updated receipt and both registered modes passed under -CPython 3.12.13. A new committed export will be checked for that addendum. +CPython 3.12.13. The addendum at `f94fc3f2e2c53bdd629a38d0a4b8a962829d43ea` passed +committed-byte and official-export checks. Both new read-only, Git-free, +empty-PATH replays passed with all 26 export files unchanged. Their common +stdout SHA256 is `b159444b16c37da899d040aca79f594acd8503da6c4ab5519a85da5c3f75ae50`. +The five separate optimized-process corruption controls passed again. +Only README scope wording and this validation prose changed afterward. The checkout is shallow. Full-history ancestry is not claimed. Every pending predecessor byte required by the proof is retained and checked directly. From 9de3f9183a242715b664ae8747a8540bf6cc38e5 Mon Sep 17 00:00:00 2001 From: Ali Romman <226877337+aliromman@users.noreply.github.com> Date: Mon, 2 Nov 2026 22:35:56 -0600 Subject: [PATCH 5/5] Distinguish the new retirement caller from the fixed-root ceiling --- research/p104-monotone-frame-ceiling/PROOF.md | 2 +- research/p104-monotone-frame-ceiling/README.md | 7 +++++-- 2 files changed, 6 insertions(+), 3 deletions(-) diff --git a/research/p104-monotone-frame-ceiling/PROOF.md b/research/p104-monotone-frame-ceiling/PROOF.md index 1ea231c077..0309926f01 100644 --- a/research/p104-monotone-frame-ceiling/PROOF.md +++ b/research/p104-monotone-frame-ceiling/PROOF.md @@ -292,7 +292,7 @@ bound are the additional cost consequence proved here. ## 7. Interpretation and limits -The latest source comparison is PR6529's reported conditional .003288299. +The pinned source comparison is PR6529's reported conditional .003288299. The ratio `.003293/.003288299-1` is less than .00143. Thus the whole stated frame-only class has less than .143 percent remaining headroom. This is an upper bound on possible improvement, not an attained construction. diff --git a/research/p104-monotone-frame-ceiling/README.md b/research/p104-monotone-frame-ceiling/README.md index 0a80abaf2e..a1554eb72b 100644 --- a/research/p104-monotone-frame-ceiling/README.md +++ b/research/p104-monotone-frame-ceiling/README.md @@ -12,7 +12,7 @@ At the 0.003293 ceiling, different leaves can also use different source words. A second result applies to **any root in the stated fresh-stage billing model**, not only this fixed root. Its favorable raw-budget search needs at most **22 leaves per binary child** for 0.003293<=kappa<=0.0034, or **21 leaves** at -the current one-percent target 0.00332118199. The proof gives a bound on each +the pinned PR6529 one-percent target 0.00332118199. The proof gives a bound on each child increment, with no bound on the number of root fields or full sites. This converts an unbounded child-size search into a finite one. It does not provide a new root or a positive invoice. See `PROOF.md`, Section 6. @@ -95,4 +95,7 @@ The canonical investigation is [Discussion5593, comment19362959](https://github. Main, priority 8 and selected6030 were checked before work and at each pivot. The general coframe, period-six caller, source-word and intermediate-retirement claims were kept separate. The overlap snapshot cannot cover unpublished work. -The user's requested large complete kappa improvement remains unmet. +The newer [PR6545](https://github.com/CrocSwap/integer-mult-bounds/pull/6545) +reports 0.00329394 by changing the caller boundary. It is outside the fixed-root +ceiling here. Applicability of Section 6 to that new caller needs a separate +invoice check. The user's requested large complete kappa improvement remains unmet.