From f77a3a3e6b8aba91d71968d7864393d1aea63917 Mon Sep 17 00:00:00 2001 From: Cinna Davis <1337isnot1337@users.noreply.github.com> Date: Thu, 22 Oct 2026 10:14:55 -0500 Subject: [PATCH 1/2] Prove the typed quotient has no recursive primitive discount --- research/typed-primitive-price/.gitattributes | 1 + .../typed-primitive-price/FROZEN-INPUTS.json | 26 ++ research/typed-primitive-price/PROOF.md | 148 +++++++++++ research/typed-primitive-price/README.md | 47 ++++ research/typed-primitive-price/RESULT.json | 90 +++++++ research/typed-primitive-price/SOURCES.json | 24 ++ research/typed-primitive-price/TASK.json | 66 +++++ research/typed-primitive-price/VERIFY.json | 20 ++ .../inputs/round3-arithmetic.json | 241 ++++++++++++++++++ .../inputs/typed-interface.md | 136 ++++++++++ .../inputs/typed-word.json.gz | Bin 0 -> 263235 bytes research/typed-primitive-price/verify.py | 178 +++++++++++++ 12 files changed, 977 insertions(+) create mode 100644 research/typed-primitive-price/.gitattributes create mode 100644 research/typed-primitive-price/FROZEN-INPUTS.json create mode 100644 research/typed-primitive-price/PROOF.md create mode 100644 research/typed-primitive-price/README.md create mode 100644 research/typed-primitive-price/RESULT.json create mode 100644 research/typed-primitive-price/SOURCES.json create mode 100644 research/typed-primitive-price/TASK.json create mode 100644 research/typed-primitive-price/VERIFY.json create mode 100644 research/typed-primitive-price/inputs/round3-arithmetic.json create mode 100644 research/typed-primitive-price/inputs/typed-interface.md create mode 100644 research/typed-primitive-price/inputs/typed-word.json.gz create mode 100644 research/typed-primitive-price/verify.py diff --git a/research/typed-primitive-price/.gitattributes b/research/typed-primitive-price/.gitattributes new file mode 100644 index 0000000000..b7823e3f34 --- /dev/null +++ b/research/typed-primitive-price/.gitattributes @@ -0,0 +1 @@ +inputs/* -text diff --git a/research/typed-primitive-price/FROZEN-INPUTS.json b/research/typed-primitive-price/FROZEN-INPUTS.json new file mode 100644 index 0000000000..29080fcb6b --- /dev/null +++ b/research/typed-primitive-price/FROZEN-INPUTS.json @@ -0,0 +1,26 @@ +[ + { + "repository": "CrocSwap/integer-mult-bounds", + "commit": "dca175fed8ebde8980635fd3b4508b8cba89a1b0", + "path": "research/native-quotient-interface/REPAIRED-WARP-WORD.json.gz", + "snapshot": "inputs/typed-word.json.gz", + "git_blob": "b22174e7a93716a394f2feb45f57a3da4331275a", + "sha256": "143b655bdc7aa719d0e3f2a527ed748c50a186997c01c2f3134dd5653d42ae7d" + }, + { + "repository": "CrocSwap/integer-mult-bounds", + "commit": "dca175fed8ebde8980635fd3b4508b8cba89a1b0", + "path": "research/native-quotient-interface/TYPED-QUOTIENT-INTERFACE.md", + "snapshot": "inputs/typed-interface.md", + "git_blob": "bf4ff1af66ecd566685b481b459b1d846b8f5edb", + "sha256": "5e48990d78d0e0b7ca2f79f24338be81e7b2fa2732c261f046d46877f2febbf3" + }, + { + "repository": "CrocSwap/integer-mult-bounds", + "commit": "1eb8da6cfb35512c0cc2533413e5ca42f48e13c6", + "path": "research/kappa-round3-warp-20261022/ARITHMETIC.json", + "snapshot": "inputs/round3-arithmetic.json", + "git_blob": "c89e009ca4a5ad316c69f0371d885d72045ba415", + "sha256": "ab10f6a4a8a4f589d4b094aac663e002c43ec44774e6bd181c51915c7fa7e521" + } +] diff --git a/research/typed-primitive-price/PROOF.md b/research/typed-primitive-price/PROOF.md new file mode 100644 index 0000000000..cb2c1e2e39 --- /dev/null +++ b/research/typed-primitive-price/PROOF.md @@ -0,0 +1,148 @@ +# The typed quotient has no recursive primitive discount + +Prepared for Cinna Davis with substantial OpenAI Codex assistance. Apache-2.0. + +## 1. Exact direct implementations of the supplied interface + +Assume the exact typed ABI of PR4654 at +`dca175fed8ebde8980635fd3b4508b8cba89a1b0`. Its 112 main pairs consist +of 56 type A and 56 type B. On each pair, Q is a symmetric rank-one +Gaussian and D a symmetric rank-seven Gaussian on disjoint axes: + +``` +A = diag(D,D) S_y(Q^-1), +B = diag(D,QD) S_y(Q^-1) = S_y(1) diag(D,QD). +``` + +The second identity is exact: both B expressions send (X,Y) to +`(DX, DX+QD Y)`. There is no order-two relation for Q in this proof. + +A direct A implementation uses two rank-seven D jobs and one rank-one +query shear. The query copies X into an owned zero temporary, applies +Q^-1 with its actual monomial normalizer, adds into Y and clears the +temporary. This retains the predecessor's owned-erasure hypothesis; it +does not erase a dirty borrowed field. A direct B uses two D jobs, one +Q job on Y and one ordinary output addition. Thus each type costs exactly +`2*7^t+1` in recursive paid moment, t=1-C. + +The raw inverse transposes have equally explicit implementations: + +``` +A^-T = diag(D^-1,D^-1) S_x(-Q^-1), +B^-T = S_x(-1) diag(D^-1,Q^-1 D^-1). +``` + +The B scalar addition is on the left and acts last. Gaussian inverses +retain their actual monomial normalizers; rank does not change. The +verifier checks all four Laurent matrix identities, with independent D +and Q symbols and every sign and inverse exponent intact. + +Across all 112 pairs, direct main execution costs **224 rank-seven jobs +and 112 rank-one jobs**, in either orientation. It uses at most one owned +query temporary at a time. The source's 408 helpers remain independent +arbitrary inputs; implementing each source-normalized helper job directly +adds its complete actual target moment H_t to the paid list. All helper +monomial normalizers are retained. No helper target is counted twice. + +This is an operator-level fallback and a comparison of recursive paid +moments. ADDs, copies, clear/erase, monomial wrappers, fixed routing and +helper normalizers have finite volume-linear work. We do not assert +identical machine costs, construct a practical multiplier, or independently +reprove the typed source's native/all-width/phase theorem. + +## 2. Strict dominance at every concave exponent + +Recounting both emitted source words and their helper endpoints gives + +``` +P = {1:2177,2:424,3:389,4:147,5:50,8:8}, +H = {3:2,6:7,7:47,8:139,9:213}. +``` + +The native minus direct recursive paid moment is + +``` +G(C) = P_t - H_t -224*7^t -112, t=1-C. +``` + +It has G(0)=8. Normalize K(C)=G(C)/7^t. The constant -224 +has derivative zero. In `(P_t-H_t-112)/7^t` the rank coefficients are + +``` +{1:2065,2:424,3:387,4:147,5:50,6:-7,7:-47,8:-131,9:-213}. +``` + +For 0<=C<1, all derivative contributions are nonnegative except the +rank-six term. The rank-one term alone contributes at least +`2065/7 * log(7)`; the rank-six negative contribution has magnitude at +most `7 log(7/6)`. Since log(7)>1 and log(7/6)<1, + +``` +K'(C) > 2065/7 -7 = 288 > 0. +``` + +The elementary log inequalities are also enclosed by the pinned exact +rational interval engine. Therefore + +``` +G(C) >= 8 * 7^(-C) > 0, 0<=C<1. +``` + +The typed native word is strictly more expensive in recursive paid moment +than the direct implementation of the *same* main ABI and helper targets, +for every exponent in the range, including rank mass at C=0. + +This is stronger than a failed controller search and different from the +A-only-path obstruction of PR4660. It applies to the local offer before +any controller is chosen. A profitable consumer using this offer could +instead use the cheaper direct implementation, with the same operator +and independent helper targets. It does **not** prove that every such +consumer or every Gaussian algorithm is impossible. + +## 3. Comparison with the actual round-three native discount + +Freeze PR4610 at `1eb8da6cfb35512c0cc2533413e5ca42f48e13c6`: + +``` +H0 = {4:28,7:7,8:47,9:326}, +P0 = {1:2201,2:412,3:510,4:127,5:40,8:8}. +``` + +Its ordinary all-A, block-eight main interface has direct paid cost +`H0_t+224*8^t+112`. Put G0=P0_t-H0_t-224*8^t-112. +Its normalized rank-eight gap G0/8^t starts at -6. The corresponding +rank-one coefficient is 2089; the only negative derivative contribution +below rank eight is -7 at rank seven. Hence + +``` +(G0/8^t)' > 2089/8 -7 = 2033/8 > 0. +``` + +Exact outward enclosures give G0<0 at C=0.0062073960 and G0>0 at +C=0.0062073961. Thus the original supplier's unique direct-price crossing +is in that interval. Above it the original offer, too, has no recursive +discount over direct main jobs and direct helpers. This is a **local price +crossing**, not an integer-multiplication exponent or a universal ceiling. + +At C=.004, the original offer still discounts the direct price by about +16.9818753 per full invocation. This is useful controller budget; the +typed quotient instead has a positive surcharge. Complete controller, +strictification, compiler, BIT, precision and outer conditions remain +necessary for any actual K claim. + +## 4. Source and verification boundary + +The source word, typed interface and round-three arithmetic are retained +byte for byte, with Git blob and SHA256 authentication. Both paid censuses +and every helper rank are recomputed from the emitted word. The source's +Clifford and free-group proofs are premises here, not newly audited. + +The registered Python replay uses exact Laurent arithmetic and outward +rational power/log enclosures. It rejects an unpaid B query, a wrong Q +inverse, missing owned charges and source/profile drift. Passing it does +not establish a new native construction, selected frontier or theorem. + +Credit Dugongue and PR4654, DaysSky's quotient source, Cao Yifan and PR4610, +Cinna Davis and the PR4440/PR4470/native lineage, zxv, Jacob Alan Sussman, +the PR4660 B-autonomy observation and all retained predecessor authors. +The direct factorization of B is credited, not rediscovered here. diff --git a/research/typed-primitive-price/README.md b/research/typed-primitive-price/README.md new file mode 100644 index 0000000000..0db207d89d --- /dev/null +++ b/research/typed-primitive-price/README.md @@ -0,0 +1,47 @@ +# The typed quotient costs more than its direct main primitives + +**No new K.** Under the exact A/B interface supplied by PR4654, the native +word's recursive paid moment is strictly greater than a direct implementation +of the same main operator and all independent helper jobs, for every +`0<=C<1`. Its surcharge is at least **8 * 7^(-C)** per invocation. + +Read [PROOF.md](PROOF.md). A direct main implementation needs 224 rank-seven +jobs and 112 rank-one jobs; all 408 helper jobs are also paid. Four exact +Laurent identities cover both types in both orientations. Every sign, +inverse power and helper normalizer is retained. This is a comparison of +recursive prices; fixed routing, scalar, ownership and native admission +obligations are stated separately. + +For comparison, PR4610's original block-eight offer has a real discount +until its unique local price crossing in +**(0.0062073960, 0.0062073961)**. This crossing is a local budget threshold, +not a new multiplication exponent or a global controller bound. + +```sh +python3.12 -B scripts/research_package.py --package research/typed-primitive-price +``` + +The standard-library replay authenticates the frozen source Git blobs and +SHA256 values, recounts both emitted paid profiles and all helper targets, +checks the operator identities, proves the monotone all-exponent bounds, +and verifies the committed [RESULT.json](RESULT.json). Original compressed +and CRLF source bytes remain unchanged. `--write` is only an authoring path. + +No selected source or shared verifier changes. No selected cold replay, +new phase/native audit, complete multiplier or independent human theorem +acceptance is claimed. The typed source remains a constructive local +interface; this result shows why its lower nominal paid profile alone +does not justify another controller construction round. + +## Open next questions + +Seek a changed native word or centre recipe that restores a genuine local +discount before searching large controllers. The original supplier still +has about 16.9818753 units of direct-price discount at C=.004; any new +controller must turn that into a positive complete moment with every idle, +query, ownership and completion cost included. + +Credit Dugongue/PR4654, DaysSky, Cao Yifan/PR4610, the PR4660 B-autonomy +observation, Cinna Davis's PR4470/controller lineage, zxv, Jacob Alan Sussman +and the retained source authors. Prepared for Cinna Davis with substantial +OpenAI Codex assistance; no subagents. Apache-2.0. diff --git a/research/typed-primitive-price/RESULT.json b/research/typed-primitive-price/RESULT.json new file mode 100644 index 0000000000..48172ad9e4 --- /dev/null +++ b/research/typed-primitive-price/RESULT.json @@ -0,0 +1,90 @@ +{ + "direct_main_paid": { + "1": 112, + "7": 224 + }, + "new_kappa": false, + "normalized_derivative_lower_bound": "288", + "original_discount_endpoint": [ + "1551849/250000000", + "62073961/10000000000" + ], + "original_gap_above": { + "lower": "0.000000609979802177861027451394", + "upper": "0.000000609979802177861027451395" + }, + "original_gap_below": { + "lower": "-0.000000157165757402536960401596", + "upper": "-0.000000157165757402536960401595" + }, + "samples": { + "0": { + "original_direct_minus_native": { + "lower": "47.999999999999999999999999999999", + "upper": "48.000000000000000000000000000001" + }, + "typed_native_minus_direct": { + "lower": "7.999999999999999999999999999999", + "upper": "8.000000000000000000000000000001" + } + }, + "1/100": { + "original_direct_minus_native": { + "lower": "-28.953945191895344577529449792159", + "upper": "-28.953945191895344577529449792158" + }, + "typed_native_minus_direct": { + "lower": "77.902661003205805150124181011979", + "upper": "77.902661003205805150124181011980" + } + }, + "1/250": { + "original_direct_minus_native": { + "lower": "16.981875310258539619945498940638", + "upper": "16.981875310258539619945498940639" + }, + "typed_native_minus_direct": { + "lower": "36.169081320591341659262855607882", + "upper": "36.169081320591341659262855607883" + } + }, + "3/1000": { + "original_direct_minus_native": { + "lower": "24.706677945572085198228228950536", + "upper": "24.706677945572085198228228950537" + }, + "typed_native_minus_direct": { + "lower": "29.152950244683753346945507106141", + "upper": "29.152950244683753346945507106142" + } + }, + "3/500": { + "original_direct_minus_native": { + "lower": "1.591451854306711455285971916073", + "upper": "1.591451854306711455285971916074" + }, + "typed_native_minus_direct": { + "lower": "50.149300571869503859726803947717", + "upper": "50.149300571869503859726803947718" + } + } + }, + "scope": "Exact typed ABI and direct helper normalization; local moment comparison only", + "status": "PASS_TYPED_DIRECT_PRIMITIVE_DOMINANCE", + "typed_H": { + "3": 2, + "6": 7, + "7": 47, + "8": 139, + "9": 213 + }, + "typed_P": { + "1": 2177, + "2": 424, + "3": 389, + "4": 147, + "5": 50, + "8": 8 + }, + "uniform_gap_lower_bound": "8 * 7^(-C), 0<=C<1" +} diff --git a/research/typed-primitive-price/SOURCES.json b/research/typed-primitive-price/SOURCES.json new file mode 100644 index 0000000000..9710f0a5a1 --- /dev/null +++ b/research/typed-primitive-price/SOURCES.json @@ -0,0 +1,24 @@ +{ + "schema_version": 1, + "base_commit": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395", + "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602", + "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72" + }, + { + "path": "research/higher-moment-strictification/verify.py", + "usage": "input", + "git_blob": "4b24a5096154736ce060474eb8c72dc64c303699", + "sha256": "0ce7e7137422d5ffeda22085c81fa88471bbe963100fdad62e659f1e3bf89326" + } + ] +} diff --git a/research/typed-primitive-price/TASK.json b/research/typed-primitive-price/TASK.json new file mode 100644 index 0000000000..f1e3854ab5 --- /dev/null +++ b/research/typed-primitive-price/TASK.json @@ -0,0 +1,66 @@ +{ + "schema_version": 1, + "task_id": "complex-new-helper-inventory", + "priority_revision": 7, + "base_commit": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51", + "original_card": { + "id": "complex-new-helper-inventory", + "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?", + "starting_points": [ + "docs/research/research-family-status.md", + "research/complex-local-word-ceiling/README.md", + "research/heterogeneous-owner-assembly/README.md", + "certificates/selected-result.json", + "research/selected-k8-unrestricted-ends/README.md" + ], + "methods": [ + "Propose a changed helper inventory and derive an optimistic exact moment.", + "Design a small completed instance and independently verify its endpoint and rank bill." + ], + "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model. 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", + "research/complex-local-word-ceiling/verify.py" + ], + "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING", + "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.", + "scope": "Positive control and planning budget only; this is not a new construction or full cold replay." + }, + "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin.", + "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.", + "allowed_variations": [ + "Narrow to one changed boundary or helper type with explicit counts.", + "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope." + ], + "scope_change_triggers": [ + "Mixing only providers already inside the pinned catalogue.", + "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.", + "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts." + ], + "overlap_terms": [ + "helper inventory", + "heterogeneous-owner", + "mutable leaver" + ] + }, + "scope_status": "changed", + "actual_scope": "Construct a direct implementation of the exact mixed A/B main interface, prove strict paid-moment dominance over the typed quotient for every exponent, and quantify the original native discount endpoint.", + "scope_change_reason": "User-directed higher-K continuation: establish whether the new mixed supplier offers an arithmetic benefit before searching larger controllers.", + "bottleneck_link": "The typed local word must offer a real moment discount over direct main jobs and direct helper normalization to justify controller exploration.", + "budget_minutes": 60, + "stop_rule": "Finish exact operator identities, source-bound profile recount, all-exponent comparison and publication; do not promote local dominance to a universal controller obstruction.", + "outcome": "completed", + "evidence": "PROOF.md, RESULT.json, source snapshots and verify.py", + "author": "Cinna Davis / 1337isnot1337, with substantial OpenAI Codex assistance", + "ai_assistance": "Substantial OpenAI Codex assistance; no subagents or independent human review.", + "open_questions": [ + { + "question": "Can a changed native word or centre recipe restore a useful main-interface discount?", + "why_it_matters": "The exact typed quotient is more expensive than direct Gaussian jobs at every exponent.", + "first_step": "Emit a source-compatible lower paid profile and compare its complete local moments with the direct fallback before controller search.", + "blocked_by": "No cheaper source-compatible centre or stronger local word established." + } + ] +} diff --git a/research/typed-primitive-price/VERIFY.json b/research/typed-primitive-price/VERIFY.json new file mode 100644 index 0000000000..1ff33ddcc9 --- /dev/null +++ b/research/typed-primitive-price/VERIFY.json @@ -0,0 +1,20 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/higher-moment-strictification/verify.py" + ], + "commands": [ + { + "argv": [ + "{python}", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 120, + "checks": "Authenticate source bytes and Git blobs; recount all paid and helper jobs; verify both forward/backward Laurent factorizations; certify all-exponent dominance and the original discount root with actual charge controls." + } + ] +} diff --git a/research/typed-primitive-price/inputs/round3-arithmetic.json b/research/typed-primitive-price/inputs/round3-arithmetic.json new file mode 100644 index 0000000000..c89e009ca4 --- /dev/null +++ b/research/typed-primitive-price/inputs/round3-arithmetic.json @@ -0,0 +1,241 @@ +{ + "compiler": { + "H": 31385200000000, + "N": "2^31385200000000", + "eta": { + "lower": "0.000000204541256425394083726654", + "upper": "0.000000204541256425394083726655" + }, + "gap": "1/700000000", + "integer_test_exponent": 44836, + "logarithm_power": 1000, + "theta": "63/64", + "uniform_saving": "1057619/350000000", + "word_materialized": false + }, + "gain_over_PR4535": "319/200000000", + "kappa": "9443/3125000", + "local_helpers": { + "4": 28, + "7": 7, + "8": 47, + "9": 326 + }, + "local_paid": { + "1": 2201, + "2": 412, + "3": 510, + "4": 127, + "5": 40, + "8": 8 + }, + "margin_at_old_root_upper": [ + "78072058023577019965454984044224523228758899702419408235/196159429230833773869868419475239575503198607639501078528", + "1249152928377232319447279744707592371660142395238823308861/3138550867693340381917894711603833208051177722232017256448" + ], + "native_receipt_sha256": "eec3b14241430b6c9f5de3cbf0e3d45a172d8071d2dccd2b59d56a1fbac3d428", + "negative_controls": { + "closed_epsilon": [ + "epsilon_below_one", + "global_gamma_power_slack", + "actual_logarithmic_gamma_guard", + "source_axis_superpolynomial", + "gaussian_above_main", + "new_semantic_guard_with_C1_one" + ], + "closed_uniform": [ + "new_complex_uniform_saving_above_b" + ], + "old_BIT": [ + "actual_bit_above_effective" + ], + "unfunded_endpoint": [ + "main_transform_above_kappa", + "global_coordinate_router_above_kappa", + "fused_gaussian_above_kappa", + "group_bit_router_above_kappa", + "boundary_algebraic_matrix_above_kappa" + ] + }, + "nonrecursive": { + "local_routes": 4802000, + "local_scalar": 158168, + "raw_scalar_upper": 181098496 + }, + "ordinary_BIT": { + "admitted": "17/5000", + "after_cover": "69/20000", + "atom": "1/250", + "before_cover": "3459/1000000", + "ordinary_saving": "8590884599/2500000000000", + "prime_rule": "odd q > 2^26*N^5 avoiding all fixed denominators and rank witnesses" + }, + "outer": { + "functional_checks": { + "actual_new_BIT_stock_sublinear": "174299999253/175000000000", + "fixed_halo_explicit_metadata_branch": "13475000001/175000000", + "new_BIT_row_stock_sublinear_in_active_group": "697899997009/700000000000", + "new_semantic_group_guard": "3/700000000" + }, + "margins": { + "boundary_algebraic_matrix": "296133039576953/98000000000000000", + "fused_gaussian": "296133038730859/98000000000000000", + "global_coordinate_router": "528809/175000000", + "group_bit_router": "370166473413573/122500000000000000", + "group_scans": "349999999/350000000", + "main_transform": "296132899576953/98000000000000000", + "microbox_face_corrections": "13475000001/175000000", + "polynomial_products": "699999999/700000000", + "sparse_union_movement": "40108633039576953/98000000000000000" + }, + "parameters": { + "a": "528809/175000000", + "b": "2115237/700000000", + "beta": "699999997/700000000", + "bit_saving": "17/5000", + "cap_xi": "89108632899576953/98000000000000000", + "complex_uniform_saving": "1057619/350000000", + "d": "1/700000000", + "epsilon": "699999999/700000000", + "kappa": "9443/3125000", + "lambda_": "199395647/200000000", + "lambda_prime": "139576953/140000000", + 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Requires separately admitted changed native programs, paid transports and absolute phases. 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Apache-2.0. + +## Sufficient constructive statement + +Starting with the credited kernel-153 complex quotient word, there is an explicit replacement of its 72 shared native address maps and explicit monomial input/output normalizers on every main port such that: + +- Every actual shared native warp has rank zero and an exact monomial realization, including its absolute phase and inverse-transpose orientation. +- The complete local paid profile is unchanged: + + P={1:2177,2:424,3:389,4:147,5:50,8:8}. + +- All 408 arbitrary independent helpers retain their actual whole-job target census: + + H={3:2,6:7,7:47,8:139,9:213}. + +- Of the 112 independent main pairs, exactly 56 realize + + F_A=D S_y(Q⁻¹), + + and exactly 56 realize + + F_B=diag(D,QD) S_y(Q⁻¹). + +Here D is the Gaussian transform on seven fresh address axes, Q is the Gaussian transform on a separate central axis, and the ninth axis is a spectator. In the field ordering X=roles 0 through 111 and Y=roles 112 through 223, the extra central Q in type B acts on Y, the target of the forward shear. The backward interface is the inverse transpose of the corresponding forward matrix. These statements hold at every digit-column fiber width f, with D and Q replaced by their f-column counterparts. + +This is a sufficient typed local interface. It establishes neither the old common rank-eight interface nor a completed improved multiplication bound. Its main purpose is to give a controller an actual proved operator to consume, instead of pricing changed output charts as though they were unchanged. + +## 1. Repair all shared native maps constructively + +Let H₈ denote the image of the nine-dimensional ambient under the kernel-153 quotient. Before spectator padding it is the coordinate eight-plane. Every native operator has a shared contract H₈→H₈ and one additional equal-dimensional pair A→B. Twelve operators also have a nonzero line a→b. In the submitted finite word the line belongs to A if and only if b belongs to B. These incidences are checked explicitly, not inferred from dimensions alone. + +For a two-subspace contract, choose bases of A and B and extend each to H₈. Mapping corresponding basis vectors gives an invertible map on H₈. For a line plus subspace, put a and b first in their respective bases if they belong to A and B; otherwise first choose bases of A and B, then append a and b. Extend both lists to bases of H₈. Map the corresponding bases, and fix the unused ninth coordinate. This constructs one invertible G for every operator, satisfying all occurrences of that operator simultaneously. Negative occurrences use its inverse. + +Use the native monomial K_G specified by the retained source. Every actual physical warp is + + C_input.adjoint().then(K_G).then(C_output), + +or uses K_G⁻¹ for a negative occurrence. The canonical-frame Lagrangian formula implies its Fourier rank is zero whenever G maps the input subspace onto the output subspace. The exact Clifford lowering constructs the monomial and checks its tableau and absolute phase. Shared-map consistency is essential: separately choosing a map at each occurrence would not preserve the arbitrary-input symbolic identity. + +The exact emitted word passes both retained literal interpreters on all 632 arbitrary-input fields in both orientations. Therefore replacing the shared operators preserves the source's formal all-column identity, with no initialized-helper assumption. All 180 actual warp classes were freshly lowered in both orientations and had exact rank zero, including absolute phases. No Fourier call is hidden inside a supposed free warp. The eight paid center operations and every helper-final repadding call are included in P. + +## 2. The exact two-field endpoint before normalization + +At every main pair, X starts in the central line c and ends in H₈. Y starts at ground and ends in a subspace A. Writing C_H, C_c and C_A for the exact canonical frame operators, the forward physical block matrix is + + [ C_H C_c⁻¹ 0 ] + M = [ C_A C_c⁻¹ C_A ]. + +There are precisely two cases: + +- Type A: dim A=7, A⊂H₈, and c∉A, so A⊕c=H₈. +- Type B: A=H₈. + +Both cases occur 56 times. These are properties of the literal source frames, independently checked for each port. The distinction is why a uniform common rank-eight replacement was invalid. + +## 3. Explicit monomial normalizers + +Choose standard central axis e₀ and standard seven-plane E=span(e₁,…,e₇). Put Q=C_e₀ and D=C_E. They act on disjoint axes, so they commute. + +In type A, choose an invertible binary map G sending e₀ to c and E to A. It consequently sends span(e₀,E) to H₈. In type B, choose G sending e₀ to c and span(e₀,E) to H₈. Extend it on the spectator coordinate. Let N=K_G, an explicit native monomial. Define + + T_Y=D in type A, + T_Y=QD in type B, + + M_Y=N, + M_X=C_c N Q⁻¹, + O_X=D Q N⁻¹ C_H⁻¹, + O_Y=T_Y N⁻¹ C_A⁻¹. + +All products use ordinary matrix multiplication, with the rightmost factor applied first. + +Each of the four M_X,M_Y,O_X,O_Y is monomial. To see this without assuming an oracle, use + + L_U=C_U(L₀)={(x,z):x∈U, x+z∈U⊥}. + +The monomial K_G sends L_U to L_GU. Every canonical frame is involutory at the symplectic level. Therefore M_X sends L₀ to L₀ because N sends L_e₀ to L_c. Likewise O_X sends L₀ to L₀ because N sends L_span(e₀,E) to L_H₈. In type A, O_Y is monomial because N sends L_E to L_A; in type B use L_span(e₀,E) instead. A Clifford preserving L₀ is monomial. Exact lowering resolves its affine permutation, quadratic phase and absolute scalar phase. + +Direct multiplication, without approximation, gives + + O_X (C_H C_c⁻¹) M_X = D, + O_Y C_A M_Y = T_Y, + O_Y (C_A C_c⁻¹) M_X = T_Y Q⁻¹. + +Thus diag(O_X,O_Y) M diag(M_X,M_Y) is exactly F_A or F_B as claimed. No independently chosen column is credited twice; the wrappers act on the original two independent main inputs and their actual outputs. Helpers remain independent and unchanged by these external main-only wrappers. + +The monomial wrappers are explicit finite nonrecursive operations. They must be counted in any subsequent finite-template route/scalar allowance; they are not zero-cost machine instructions. Their degree, coordinate dimension and coefficients are fixed, so they contribute fixed linear per-record work under the native lowering. All their phases have modulus one, so the wrappers do not introduce a growing conditioning loss. + +## 4. Inverse transpose and every fiber width + +The actual backward source word uses the inverse-transpose logical identity and the corresponding conjugated frame operators. If F=O M N is the forward identity, then + + F⁻ᵀ=O⁻ᵀ M⁻ᵀ N⁻ᵀ. + +The inverse transpose of a monomial is monomial. Hence the normalized backward type is exactly F_A⁻ᵀ or F_B⁻ᵀ, rather than an independently guessed orientation. The extra central Q is on physical Y before taking this dual; its exponent changes sign under inverse transpose. + +For fiber width f, tensor each ADDRESS-OPERATOR block identity f times. For example, from O_X A_X M_X=D one obtains O_X^(⊗f) A_X^(⊗f) M_X^(⊗f)=D^(⊗f). Apply the same argument separately to the Y diagonal and the off-diagonal coefficient. This preserves the two-field block shape and scalar coefficient 1. It does not tensor the entire 2×2 field matrix, which would create additional field rows. Monomial tensor powers remain explicit monomials, and the chronological arbitrary-input identity at every fiber width follows from the independent free-group-algebra argument below. + +## 5. A universal free-group-algebra evaluation proof + +The logical endpoint is not inferred by tensoring a numerical one-column field matrix. The retained literal interpreter computes a stronger symbolic identity. + +Let F be the free group on 72 named generators g₁,…,g₇₂ and R=Q[F] its group algebra. A row is a finitely supported map from pairs (initial field, reduced group word) to rational coefficients. Initialize each of the 632 persistent fields as its own basis vector. The owned C field is also represented symbolically until it is explicitly cleared. Process every record as follows: + +- A native warp with signed operator i multiplies each coefficient on the LEFT by g_|i| or its inverse. The implementation prepends the signed symbol and removes it only when it cancels the adjacent inverse symbol. +- ADD and SCALE perform exact rational module arithmetic. COPY duplicates the source expression only into permitted owned storage. CLEAR sets only permitted owned storage to zero. +- A paid frame change is the identity on logical values; its physical conjugating frame operators are accounted separately. Phase and adapter markers do not modify the logical row. + +These are precisely module operations in the free R-module on the initial field symbols. Induction on the record index therefore proves that the interpreter's row is the corresponding formal logical expression, before any native operator has been assigned a numerical matrix. In particular, no relation such as commutation, finite order, dimension, a chosen binary G map, or a one-column coincidence is used. + +At termination, the checked forward rows are X_i↦X_i, Y_i↦Y_i+X_i and h_j↦h_j; the backward rows are X_i↦X_i−Y_i, Y_i↦Y_i and h_j↦h_j. Every surviving coefficient has the EMPTY group word. The owned C row is zero. The fresh receipt records 632 persistent rows, 744 exact terms and zero nonempty group words in each orientation. Changing one occurrence to a new unshared generator produces a nonzero endpoint difference and is explicitly rejected. + +Now take any vector space V over Q or C and assign ANY invertible linear operator T_i∈GL(V) to each shared generator. Send a reduced word to the ordered product of those operators and extend Q-linearly. This is a well-defined algebra homomorphism R→End(V): the only reduction used, g_i g_i⁻¹=1, is valid because T_i is invertible. Extending it componentwise to module rows sends every step of the formal computation to the actual vector-valued operation. Hence the final formal identity proves the same endpoint for arbitrary independent V-valued inputs. + +Apply this evaluation independently at every fiber width f, assigning the explicitly constructed shared K_G,f and their inverses. The same operator must be used at every occurrence of its name; this is why the simultaneous shared-map construction in section 1 was required. The 72 new maps satisfy that condition. The backward interpreter uses a separately named set of generators, so the result also applies to the inverse-transpose assignments. This proves the changed word's universal logical all-fiber endpoint directly. The only tensor arguments needed are for individual frame/normalizer operator identities, where tensor powers do preserve composition; no assertion that tensor powers preserve sums is made. + +This evaluation theorem is completely separate from a contraction or a complete multiplication algorithm. It proves the arbitrary-input operator identity and does not supply a favorable paid invoice by itself. + +## 6. Controller relevance and paid obligations + +The type-B identity has a useful exact rewriting: + + diag(1,Q) S_y(Q⁻¹)=S_y(1) diag(1,Q). + +Also Q² is monomial. This permits central-parity row gauges that can change whether a query coefficient requires a rank-one Gaussian call. It does not make all queries free: an ADD across different central gauges can itself require a paid rank-one call. A complete controller must charge every such bridge and derive its final central target parities from the actual diagonal operator. The seven-axis target ranks cannot be priced with the old eight-axis invoice. + +The receipt intentionally states only a typed local supplier. A full contraction, compiler, native caller and larger κ must be proved using the resulting complete paid and productive profiles. + +## Evidence, source status and credit + +`repair_shared_warps.py` constructs the repaired 72 shared maps, resynthesizes the word, performs two-interpreter arbitrary-input replay and lowers all actual warp classes in both orientations. Its word payload SHA-256 is `599da9a45213bc4cd98d897d6bd00e5e4bc2af8160e90ec50aec467e0c6236eb`. + +`typed_pair_normalizers.py` constructs all 448 monomial wrappers, lowers them with absolute phases and checks all 336 block identities by exact tableaux and exact zero-state vectors. Its normalizer payload SHA-256 is `7617a2893f7cb117beb35c499a02feac3d2b0a6b3e60bdd4aa198a680a26ab54`. The check takes about 18 seconds with CPython 3.12. These are exact finite certificates plus the all-fiber argument above, not Lean verification or an independent human review. + +Credit DaysSky for the kernel quotient and helper repadding in #4596/#4629/#4633; Nick Petalas for the quotient family classification in #4627; Ryanson Jonathan for the endpoint-frame audit in #4604; Cinna Davis and the PR #4470 native/helper word; Cao Yifan for the shared-generator interface work in #4610; Dugongue for the transported/arbitrary-frame and native Word RAM work; zxv and Jacob Alan Sussman for retained native and compiler/matching results; and all original source notices. 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m%Kd*#{LMQ?{vWHTq_5GE$Sc~1b;vE`F%;)KJxAcxqyGn@Lx)cQ literal 0 HcmV?d00001 diff --git a/research/typed-primitive-price/verify.py b/research/typed-primitive-price/verify.py new file mode 100644 index 0000000000..5db42ebc2e --- /dev/null +++ b/research/typed-primitive-price/verify.py @@ -0,0 +1,178 @@ +#!/usr/bin/env python3 +"""Source-bound primitive-price comparison. Apache-2.0; Codex assistance.""" +import argparse +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha1, sha256 +import gzip +import importlib.util +import json +from pathlib import Path + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def add(*values): + return tuple(sum(v[i] for v in values) for i in (0, 1)) + + +def authenticate(): + for row in json.loads((HERE/'SOURCES.json').read_text())['files']: + if row['usage'] == 'input': + need(sha256((ROOT/row['path']).read_bytes()).hexdigest() == row['sha256'], 'Live interval engine pin') + for row in json.loads((HERE/'FROZEN-INPUTS.json').read_text()): + raw = (HERE/row['snapshot']).read_bytes() + need(sha256(raw).hexdigest() == row['sha256'], 'Frozen source bytes') + need(sha1(b'blob '+str(len(raw)).encode()+b'\0'+raw).hexdigest() == row['git_blob'], 'Frozen Git blob') + + +def engine(): + path = ROOT/'research/higher-moment-strictification/verify.py' + spec = importlib.util.spec_from_file_location('primitive_price_intervals', path) + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + return module + + +def typed_profiles(): + word = json.loads(gzip.decompress((HERE/'inputs/typed-word.json.gz').read_bytes())) + c = word['canonical'] + H = Counter() + for role in range(224, 632): + a, b = (c['frames'][c[key][role]] for key in ('start','final')) + def span(basis): + s = {0} + for x in basis: + s |= {v^x for v in tuple(s)} + return s + need(span(a) <= span(b), 'Nested arbitrary helper interface') + H[len(b)-len(a)] += 1 + profiles = [] + for direction in ('forward','backward'): + P = Counter() + for event in word['programs'][direction]: + if event[0] in ('climb','center','cache_child'): + rank = event[-1] + old, new = event[-3:-1] + need(rank == abs(len(c['frames'][new])-len(c['frames'][old])), 'Emitted paid rank') + P[rank] += 1 + profiles.append(P) + need(profiles[0] == profiles[1], 'Both emitted paid censuses') + need(H == {3:2,6:7,7:47,8:139,9:213}, 'Typed helper census') + need(profiles[0] == {1:2177,2:424,3:389,4:147,5:50,8:8}, 'Typed paid census') + return H, profiles[0] + + +# Exact Laurent polynomials, with commuting symbols D and Q. No finite order. +def plus(a, b): + out = dict(a) + for key, value in b.items(): + out[key] = out.get(key, 0)+value + return {k:v for k,v in out.items() if v} + + +def times(a, b): + out = {} + for (d,q), x in a.items(): + for (e,r), y in b.items(): + k = (d+e,q+r) + out[k] = out.get(k, 0)+x*y + return {k:v for k,v in out.items() if v} + + +def mm(a, b): + return [[plus(times(a[i][0],b[0][j]),times(a[i][1],b[1][j])) for j in range(2)] for i in range(2)] + + +def mono(d=0,q=0,c=1): + return {(d,q):c} + + +def operator_identities(): + one, zero, d, dq = mono(), {}, mono(1), mono(1,1) + Sy = [[one,zero],[mono(q=-1),one]] + A = [[d,zero],[mono(1,-1),d]] + B = [[d,zero],[d,dq]] + need(mm([[d,zero],[zero,d]], Sy) == A, 'Direct forward type A') + need(mm([[one,zero],[one,one]], [[d,zero],[zero,dq]]) == B, 'Direct forward type B') + # The direct inverse transposes are literal transposes of inverses. + Adual = [[mono(-1),mono(-1,-1,-1)],[zero,mono(-1)]] + Bdual = [[mono(-1),mono(-1,-1,-1)],[zero,mono(-1,-1)]] + Ax = [[one,mono(q=-1,c=-1)],[zero,one]] + Bx = [[one,mono(c=-1)],[zero,one]] + need(mm([[mono(-1),zero],[zero,mono(-1)]], Ax) == Adual, 'Direct raw backward type A') + need(mm(Bx, [[mono(-1),zero],[zero,mono(-1,-1)]]) == Bdual, 'Direct raw backward type B') + # Omitting Q from the B diagonal, or replacing Q^-1 by Q, is false. + need(mm([[one,zero],[one,one]], [[d,zero],[zero,d]]) != B, 'B query is actually paid') + need(mm([[d,zero],[zero,d]], [[one,zero],[mono(q=1),one]]) != A, 'Q inverse control') + + +def gap(E, H, P, block, saving): + t = 1-saving + return add(E.moment(P,t), E.scale(E.moment(H,t),-1), + E.scale(E.power(block,t),-224), (Q(-112),)*2) + + +def certify(): + authenticate() + E = engine() + H, P = typed_profiles() + operator_identities() + # Normalized derivative lower bounds in PROOF.md retain just rank one. + log7, log8 = E.log_bounds(7), E.log_bounds(8) + log76 = add(log7,E.scale(E.log_bounds(6),-1)) + log87 = add(log8,E.scale(log7,-1)) + need(log7[0]>1 and log8[0]>1 and log76[1]<1 and log87[1]<1, 'Elementary log bounds') + need(Q(2065,7)-7 == 288 > 0, 'Typed normalized gap increases for all C') + mass = lambda p:sum(r*n for r,n in p.items()) + need(mass(P)-mass(H)-224*7-112 == 8, 'Every direct helper and central job charged') + source = json.loads((HERE/'inputs/round3-arithmetic.json').read_bytes()) + Ho = {int(k):n for k,n in source['local_helpers'].items()} + Po = {int(k):n for k,n in source['local_paid'].items()} + need(Ho == {4:28,7:7,8:47,9:326}, 'Round-three helper input') + need(Po == {1:2201,2:412,3:510,4:127,5:40,8:8}, 'Round-three paid input') + need(Q(2089,8)-7 == Q(2033,8)>0, 'Original normalized gap strictly increases for all C') + lo, hi = Q('0.0062073960'), Q('0.0062073961') + below, above = gap(E,Ho,Po,8,lo),gap(E,Ho,Po,8,hi) + need(below[1]<0 and above[0]>0, 'Unique original discount endpoint') + samples = {} + for C in (Q(0),Q('0.003'),Q('0.004'),Q('0.006'),Q('0.01')): + G = gap(E,H,P,7,C) + lower = E.scale(E.power(7,-C),8) + need(G[0]>=lower[1] if C else G[0]<=8<=G[1], 'Uniform typed dominance lower bound') + samples[str(C)] = {'typed_native_minus_direct':E.display(G), + 'original_direct_minus_native':E.display(E.scale(gap(E,Ho,Po,8,C),-1))} + # An actually removed centre reverses the integer boundary; this is a false invoice. + unpaid = dict(P) + unpaid[8] -= 2 + need(mass(unpaid)-mass(H)-224*7-112 < 0, 'Unpaid owned jobs control') + return {'status':'PASS_TYPED_DIRECT_PRIMITIVE_DOMINANCE', + 'scope':'Exact typed ABI and direct helper normalization; local moment comparison only', + 'typed_H':H,'typed_P':P,'direct_main_paid':{'1':112,'7':224}, + 'uniform_gap_lower_bound':'8 * 7^(-C), 0<=C<1', + 'normalized_derivative_lower_bound':'288', + 'original_discount_endpoint':[str(lo),str(hi)], + 'original_gap_below':E.display(below),'original_gap_above':E.display(above), + 'samples':samples,'new_kappa':False} + + +def main(): + p = argparse.ArgumentParser() + p.add_argument('--write',action='store_true') + args = p.parse_args() + result = json.loads(json.dumps(certify())) + if args.write: + (HERE/'RESULT.json').write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + else: + need(json.loads((HERE/'RESULT.json').read_text()) == result, 'Committed exact receipt') + print(json.dumps(result,sort_keys=True)) + + +if __name__ == '__main__': + main() From 84cf25a580db9ed2b4b1371c40689a035711364e Mon Sep 17 00:00:00 2001 From: Cinna Davis <1337isnot1337@users.noreply.github.com> Date: Thu, 22 Oct 2026 10:24:45 -0500 Subject: [PATCH 2/2] State the direct helper child-rank and compiler boundary --- research/typed-primitive-price/PROOF.md | 8 ++++++++ research/typed-primitive-price/README.md | 5 +++++ 2 files changed, 13 insertions(+) diff --git a/research/typed-primitive-price/PROOF.md b/research/typed-primitive-price/PROOF.md index cb2c1e2e39..86b640dd4e 100644 --- a/research/typed-primitive-price/PROOF.md +++ b/research/typed-primitive-price/PROOF.md @@ -44,6 +44,14 @@ arbitrary inputs; implementing each source-normalized helper job directly adds its complete actual target moment H_t to the paid list. All helper monomial normalizers are retained. No helper target is counted twice. +Some direct helper jobs have rank nine. The fallback therefore does not +preserve the native word's rank-eight local child cap. The price comparison +is not drop-in compiler admission: any replacement must recheck child ranks, +strictification tails and allocation. If a consumer introduces these helpers +only as fresh independent bystanders, its direct main implementation can +instead omit both their inputs/targets and their matching direct paid jobs; +that cancellation preserves the surplus, but changes the caller's inventory. + This is an operator-level fallback and a comparison of recursive paid moments. ADDs, copies, clear/erase, monomial wrappers, fixed routing and helper normalizers have finite volume-linear work. We do not assert diff --git a/research/typed-primitive-price/README.md b/research/typed-primitive-price/README.md index 0db207d89d..6828a89cdd 100644 --- a/research/typed-primitive-price/README.md +++ b/research/typed-primitive-price/README.md @@ -12,6 +12,11 @@ inverse power and helper normalizer is retained. This is a comparison of recursive prices; fixed routing, scalar, ownership and native admission obligations are stated separately. +The direct helper implementation includes rank-nine jobs, so it does not +retain the native word's rank-eight child cap. Its use in a compiler needs +new rank/tail checks, or a caller that removes neutral fresh helper jobs +from both target and paid lists. Local price dominance alone is not admission. + For comparison, PR4610's original block-eight offer has a real discount until its unique local price crossing in **(0.0062073960, 0.0062073961)**. This crossing is a local budget threshold,