From ebe84d415fac7f92a10c20b7a6c3dd95ce859bfb Mon Sep 17 00:00:00 2001 From: Ali Romman <226877337+aliromman@users.noreply.github.com> Date: Mon, 26 Oct 2026 17:14:00 -0500 Subject: [PATCH 1/3] Retain exact active-mode predecessor records --- .../.gitattributes | 1 + .../ORIGINS.json | 23 + .../source/PR5745-PROOF.md | 160 +++ .../source/PR5755-RANK1.json | 1106 +++++++++++++++++ .../source/PR5755-README.md | 90 ++ 5 files changed, 1380 insertions(+) create mode 100644 research/active-mixed-rank-obstruction/.gitattributes create mode 100644 research/active-mixed-rank-obstruction/ORIGINS.json create mode 100644 research/active-mixed-rank-obstruction/source/PR5745-PROOF.md create mode 100644 research/active-mixed-rank-obstruction/source/PR5755-RANK1.json create mode 100644 research/active-mixed-rank-obstruction/source/PR5755-README.md diff --git a/research/active-mixed-rank-obstruction/.gitattributes b/research/active-mixed-rank-obstruction/.gitattributes new file mode 100644 index 0000000000..7467454cb8 --- /dev/null +++ b/research/active-mixed-rank-obstruction/.gitattributes @@ -0,0 +1 @@ +source/** -text diff --git a/research/active-mixed-rank-obstruction/ORIGINS.json b/research/active-mixed-rank-obstruction/ORIGINS.json new file mode 100644 index 0000000000..34ca7781bf --- /dev/null +++ b/research/active-mixed-rank-obstruction/ORIGINS.json @@ -0,0 +1,23 @@ +[ + { + "local": "source/PR5755-README.md", + "head": "b66de876c724361b49f9dfe5ef70016862caab09", + "path": "research/active-rank1-controller/README.md", + "git_blob": "d70220ba19580ee046350ec60ac3f39512a1f8a5", + "sha256": "bd010d3ed1e405b98ac7e712a181e05282c8b89366cbf39b5420c830595b21c9" + }, + { + "local": "source/PR5755-RANK1.json", + "head": "b66de876c724361b49f9dfe5ef70016862caab09", + "path": "research/active-rank1-controller/rank1-controller.json", + "git_blob": "493511ae3f4df44cc1afb4407625dfe5820bc99a", + "sha256": "3b5ef134555e4bd3ef2f0ba1939f19f2b44164d420dabec40c55cdf5fde28ce2" + }, + { + "local": "source/PR5745-PROOF.md", + "head": "e82b64e91e57c838ff2028bcf2219203966bf9ed", + "path": "research/active-mode-delivery-frontier/PROOF.md", + "git_blob": "54ce2c79bbcfbe722d15c87bc2634e582284903d", + "sha256": "fd16bfbf4279f3988add6c2a1ffb5bee134b6ec838e7a084f0f92d34244aeaa3" + } +] diff --git a/research/active-mixed-rank-obstruction/source/PR5745-PROOF.md b/research/active-mixed-rank-obstruction/source/PR5745-PROOF.md new file mode 100644 index 0000000000..54ce2c79bb --- /dev/null +++ b/research/active-mixed-rank-obstruction/source/PR5745-PROOF.md @@ -0,0 +1,160 @@ +# Sharp paid cost of canonical active-mode delivery + +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. + +## 1. Fixed source and delivery contract + +Use PR5720's exact F9-SOURCE-DAG-ALL8 backward word. Omit only its explicit +unimplemented physical_add tail. The 632 independent inputs comprise 112 X, +112 Y and 408 helpers. Owned scratch starts at zero. The eight mode rows R +satisfy RR^T=56I. Their captured payloads are c_j=R_j Y/4. A complete repair +changes the main scalar endpoint from X-AY to X-Y, where A=I-R^T R/4, +and preserves every Y and helper target. + +A channel takes one scalar linear combination of these original mode arrays. +It uses a paid canonical transport from ground frame 0 to one frame g. It can +then add scalar multiples of that payload to any currently available fields +in frame g, at any number of times. The payload can remain in owned storage +between those times. All such control times use the unchanged source word; +the payload is not recomputed from a changed live field. Its normalized value +is constant in that frame. A different frame, address word, extra query, +changed source word, or feedback-dependent payload is outside this contract. + +A rank-r canonical transport costs H_r, with H_r positive and nondecreasing +for r>=1. The actual rank is the Clifford Gaussian grade, checked below for +each attaining transport. The theorem grants free arbitrary rational scatter +coefficients, arbitrary payload availability times, and free final correction +of every helper whose original start frame is ground. These relaxations can +only help a proposed delivery. Native and ordinary implementation costs are +not made free in the attaining word. + +A channel is clean if its endpoint, after quotienting those helper outputs, +is a linear combination of the eight desired main mode vectors and has zero +other main or Y output. The complete delivered correction has rank eight in +this output quotient because R has eight independent rows. + +## 2. Complete response spaces + +For a cut t and current field d, let v_(t,d) be the scalar endpoint response +to a unit additive impulse at that field. Propagate endpoint columns backward +through the literal word. For an ADD d+=a s, the source column becomes +v_s+a v_d. For COPY d=s it becomes v_s+v_d and v_d becomes zero. CLEAR makes +v_d zero; SCALE multiplies it. Frame changes update the set of available +fields. Native warp words are evaluated on their common fixed vector as in +PR5720. This is a necessary scalar quotient, not a new physical theorem. + +At frame g and time t, the possible scatter response is the span of all +v_(t,d) with current frame g. An invertible coframed ADD or nonzero SCALE +preserves that entire span. Thus there is no need to repeat elimination at +such an event. The replay still updates every column and checks each contact. +It checks every other span change and every final frame. Identical complete +field-column lists may share a result. Zero columns cannot add a direction. + +For each column, separate its main part into an R^T component and a nuisance +component by a fixed rational left inverse of R^T. Keep all Y coordinates and +all helper coordinates whose original start is not ground in the nuisance. +The ground-helper coordinates are the explicitly granted free quotient. +Eliminate nuisance columns while retaining mode components and exact control +witnesses. Each dependence supplies one pure-mode vector. Reducing those +vectors gives exactly the attainable mode subspace: these are the standard +column-kernel image operations, not a bounded coefficient search. + +The complete source has 5,331 distinct checked frame states and 3,963 distinct +projected endpoint columns. Only eight rank-eight frames have a nonzero clean +response before full rank. They give 41 frame states and 41 response vectors. +The final full frame gives all eight modes. No rank below eight has a clean +response. The eight frames are 254, 274, 279, 280, 281, 619, 621 and 624. +All positive witnesses are retained in CUTS.json. + +Temporal reuse is checked separately within the same pass. For each frame g, +collect the span of its response columns over every time, not just one cut. +The elimination processes 5,715 distinct frame/column pairs. There is still no +clean response at rank below eight. The result has one mode direction at each +of the eight rank-eight frames and eight at the terminal full frame. These +sixteen exact witnesses are in TEMPORAL.json. Rank-nine histories need not be +expanded further: the terminal state already provides the entire eight-mode +quotient, and no rank-nine channel costs less than H8. + +This temporal check is essential. A single-time exclusion alone would not +exclude reuse of one paid payload at several times. The positive control has +two noisy responses whose difference creates a clean mode across two times. + +## 3. Sharp tariff theorem + +First suppose every channel is clean. The source calculation proves that +each nonzero channel has r>=8. At least eight channels are required to produce +a rank-eight mode correction, because one scalar payload has mode rank at +most one. Therefore the total canonical delivery cost is at least 8H8. +This remains true for any number of clean channels and arbitrary temporal reuse. + +The same lower bound holds without the clean-channel requirement if there +are at most eight channels. Write the full output correction as VB, where +each column of V is one channel response and B contains its payload +coefficients on the eight independent modes. The desired correction has +rank eight. Thus there must be exactly eight channels, V has rank eight, +and its column space equals the desired mode subspace. Each column is then +clean in the stated quotient. The first argument applies. + +Eight explicit rank-eight deliveries attain the bound. Thus 8H8 is the exact +minimum in these stated models for every positive nondecreasing tariff. +The original removed center allowance is also 8H8. No recursive-paid saving +is available from these models. Nine or more jointly noisy channels can have +a larger response span whose nuisance cancels only after combining payloads; +the rank argument does not exclude that model. Nor does it exclude changes +to the native frame/address contract or to the underlying word. + +## 4. A legal complete attainment + +For each mode, use the earliest retained rank-eight single-field witness. +The source already holds c_j in an owned ground field C(j+1). Remove its +original helper-precompensation ADDs. At the witness cut, transport that owned +field from ground to the specified frame, subtract it from the witness field, +and CLEAR the owned field. No arbitrary helper is overwritten or erased. +The target fields, one for each mode, are bound in RESULT.json. + +The new transport is the conjugate canonical frame map for the backward word. +The retained exact compiler lowers it to a rank-eight Gaussian child with +ordinary monomial wrappers. The compact phase-sensitive engine checks the +entire tableau and absolute phase, and a separate literal zero-column check +agrees. A global-phase mutant is rejected for each of the eight maps. + +Replay the complete program with all 632 original columns and the original +noncommuting native warp words. Only adjacent inverse words cancel. The result +is X-Y on the main rows, Y unchanged, all 408 helper identities, the original +final frames, and zero on all nine owned fields. All scalar ADD contacts are +coframed. Omitting a paid transport is rejected as a coframe mismatch; restoring +the removed helper debit or reversing an injection sign fails the complete +endpoint test. + +The raw construction also contains operations on proved zero owned data. +Remove only zero-source ADDs, zero SCALEs, zero-to-zero COPYs, and redundant +CLEARs that do not change a frame. Replay the emitted word again. This removes +eight ADDs and eight CLEARs. WORD.json contains the complete resulting word. +The original source word also receives a full independent-input replay. + +The paid profiles agree exactly: + + 2201 H1 + 412 H2 + 510 H3 + 127 H4 + 40 H5 + 8 H8. + +The completed direct-delivery word has 16,578 ADDs, 32 CLEARs and 1,164 SCALEs, +with peak nine live owned fields. The original has 16,130 ADDs, 16 CLEARs and +1,164 SCALEs, with peak one. Thus this concrete completion is dominated by +the original: 448 extra ADDs, sixteen extra CLEARs and eight extra live owned +fields, with no paid gain. It is a legal witness for the paid optimum, not an +improved implementation. The original center implementation is already the +better upper bound among these two words. + +## 5. Boundaries of the result + +The mathematical lower bound uses the scalar necessary quotient and the fixed +canonical delivery contract. It is not a lower bound for general active feedback. +The positive program adds exact new native transports but inherits the frozen +source's existing native implementation contracts. The formal-word endpoint +check retains all original inputs and warp words; it is not a fresh execution +of every historical physical wrapper or a complete multiplication caller. + +No compiler, outer theorem, finite precision, stock or tape admission is +claimed. The paid result is sharp within the model; a different eight-channel +word could have different ordinary cost. The explicit direct completion here +is worse than the original. The selected frontier is unchanged. 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"artificial_chart_weight": "50", + "reason": "a chart covers target weight greater than one", + "rejected": true + }, + "dropped_final_Y_restoration": { + "raw_chart_114_weight": "50", + "reason": "a chart covers target weight greater than one", + "rejected": true + } + }, + "source": { + "R_sha256": "7b33b6dfb494c6eafe6d87a08d6a2c2e0681e56429495b2b27949040f0f63cd1", + "direction": "backward", + "physical_add": "omitted", + "word_sha256": "934fea11e51f674ce2031b7d3127b994a921a9d83317b00359292752563a1775" + } +} diff --git a/research/active-mixed-rank-obstruction/source/PR5755-README.md b/research/active-mixed-rank-obstruction/source/PR5755-README.md new file mode 100644 index 0000000000..d70220ba19 --- /dev/null +++ b/research/active-mixed-rank-obstruction/source/PR5755-README.md @@ -0,0 +1,90 @@ +# Fixed rank-one channels cannot cheaply repair the retained modes + +This source-bound necessary-quotient result restricts the rank-one-only part of +[Ali Romman's PR5745](https://github.com/CrocSwap/integer-mult-bounds/pull/5745) +open question about nine or more jointly noisy channels. It builds on +[flyingviper's PR5720](https://github.com/CrocSwap/integer-mult-bounds/pull/5720). +No new multiplication bound or physical construction is claimed. + +## Contract + +Keep the frozen F9-SOURCE-DAG-ALL8 backward word and omit its unimplemented +physical_add tail. Each channel carries a fixed scalar combination of the eight +original captured modes, prepared in one fixed rank-one chart. It may scatter +at any number of chronological contacts in that chart. Arbitrarily many +channels and joint cancellation of their nuisance outputs are allowed. Each +channel must pay its preparation; gathering, storage and ordinary scatter costs +are optimistically ignored. Ground-helper cleanup is granted. Y must return +correctly through the stated controls: extra delivery to its final charts is +not granted for free. + +Changing the native word, recomputing a payload from changed live data, +inter-chart transport or reframing, and extra paid cleanup are outside this +contract. Native address operators use the retained common-fixed-vector +specialization. A negative necessary test is not a general feedback theorem. + +## Certificate and proof + +The retained endpoint is X-CY, Y, H, with C=I-R^T R/4. For every control form +Z=deltaX+C deltaY. If all Y outputs are restored, the desired X correction must +be supplied by Z. This allows cancellations between different channels rather +than requiring each channel to be clean independently. + +The exact census covers every response-changing interval in each actual +rank-one chart. For each chart, retain the union of possible nonzero Z output +coordinates. Assign nonnegative weights to required output coordinates such +that every chart's union has total weight at most one. Any successful set of +charts must cover every required coordinate, so its size is at least the total +weight. This argument permits arbitrary rational coefficients and cancellation; +it is not a bounded coefficient search. + +The retained certificate establishes a weight of 50 for mode5. For the full +eight-mode correction, all 112 X rows are required. The only chart reaching +multiple Z coordinates reaches twelve; give those coordinates weight 1/12 and +the other hundred weight one. Every chart has load at most one, giving a bound +of 101 distinct fixed rank-one charts and hence at least 101 channels. +For the recursive power tariff H_r=r^q H_1, 0 Date: Mon, 26 Oct 2026 17:18:45 -0500 Subject: [PATCH 2/3] Exclude arbitrary mixed-rank channels below rank eight --- .../CERTIFICATES.json | 34247 ++++++++++++++++ .../COVER-LIMIT.json | 23 + .../DISCOVERY.md | 26 + .../active-mixed-rank-obstruction/PROOF.md | 147 + .../active-mixed-rank-obstruction/README.md | 58 + .../active-mixed-rank-obstruction/RESULT.json | 105 + .../SOURCES.json | 68 + .../active-mixed-rank-obstruction/TASK.json | 124 + .../VALIDATION.md | 19 + .../active-mixed-rank-obstruction/VERIFY.json | 30 + .../responses.py | 177 + .../active-mixed-rank-obstruction/verify.py | 148 + 12 files changed, 35172 insertions(+) create mode 100644 research/active-mixed-rank-obstruction/CERTIFICATES.json create mode 100644 research/active-mixed-rank-obstruction/COVER-LIMIT.json create mode 100644 research/active-mixed-rank-obstruction/DISCOVERY.md create mode 100644 research/active-mixed-rank-obstruction/PROOF.md create mode 100644 research/active-mixed-rank-obstruction/README.md create mode 100644 research/active-mixed-rank-obstruction/RESULT.json create mode 100644 research/active-mixed-rank-obstruction/SOURCES.json create mode 100644 research/active-mixed-rank-obstruction/TASK.json create mode 100644 research/active-mixed-rank-obstruction/VALIDATION.md create mode 100644 research/active-mixed-rank-obstruction/VERIFY.json create mode 100644 research/active-mixed-rank-obstruction/responses.py create mode 100644 research/active-mixed-rank-obstruction/verify.py diff --git a/research/active-mixed-rank-obstruction/CERTIFICATES.json b/research/active-mixed-rank-obstruction/CERTIFICATES.json new file mode 100644 index 0000000000..6ecb82a5a5 --- /dev/null +++ b/research/active-mixed-rank-obstruction/CERTIFICATES.json @@ -0,0 +1,34247 @@ +{ + "rank_cuts": [ + { + "rank_cap": 5, + 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"1/3", + "919": "1/9", + "920": "1/9", + "925": "1/9", + "933": "1/9", + "934": "1/9", + "937": "1/9", + "945": "1/9", + "967": "1/9", + "1166": "4/9", + "1168": "1/3", + "1170": "1/9", + "1172": "1/3", + "1174": "1/9", + "1175": "1/9", + "1176": "1/3", + "1185": "2/9", + "1186": "1/9", + "1922": "1/9", + "1933": "2/9" + }, + "optimum": "467/9" + } + ] +} diff --git a/research/active-mixed-rank-obstruction/COVER-LIMIT.json b/research/active-mixed-rank-obstruction/COVER-LIMIT.json new file mode 100644 index 0000000000..d6dfdcdc54 --- /dev/null +++ b/research/active-mixed-rank-obstruction/COVER-LIMIT.json @@ -0,0 +1,23 @@ +{ + "frames": [ + 292, + 295, + 299, + 301, + 303, + 304, + 305, + 307, + 308, + 309, + 310, + 577, + 578, + 591, + 592, + 611, + 612 + ], + "rank_mass": 62, + "missing_target_rank": 5 +} diff --git a/research/active-mixed-rank-obstruction/DISCOVERY.md b/research/active-mixed-rank-obstruction/DISCOVERY.md new file mode 100644 index 0000000000..43e4b8a1e1 --- /dev/null +++ b/research/active-mixed-rank-obstruction/DISCOVERY.md @@ -0,0 +1,26 @@ +# Bounded discovery and final certificate + +The exact backward response census was the first test. Support-cover LPs +were used only to find candidate rational witnesses. They gave no exclusion +at maximum rank four: an integral cost-62 cover exists. Its response span +misses five target directions. + +Quotient-rank cuts then replaced the coordinate-only test. Greedy spans of +actual response spaces supplied candidate cuts. SciPy/HiGHS solved finite +relaxations during discovery. Each retained nonnegative dual weight was +rounded down to a multiple of 1/100000000. The final checker recalculates +every cut and proves every capacity inequality with exact fractions. +Floating-point success is not used as mathematical evidence. + +The search used twenty refinement rounds at maximum rank four and thirty +at maximum rank five. The stronger rank-five certificate also covers ranks +one through four. Its retained nonzero cuts seed the rank-six certificate; +those seed rank seven. Both extensions have a strict bound above 64. +This reaches the largest contiguous rank range below the known rank-eight +attainment. The earlier rank-four checkpoint is not a separate result. + +The same method was also tried on all ranks with at least nine channels. +The bounded relaxation remained below the needed price and was stopped. +There is no all-rank exclusion and no positive construction from that run. +The final package retains only checked exact certificates and the useful +cost-62 failure witness. No claim of an optimized true minimum is made. diff --git a/research/active-mixed-rank-obstruction/PROOF.md b/research/active-mixed-rank-obstruction/PROOF.md new file mode 100644 index 0000000000..b85d841bec --- /dev/null +++ b/research/active-mixed-rank-obstruction/PROOF.md @@ -0,0 +1,147 @@ +# A mixed-rank obstruction for immutable mode delivery + +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. + +## Contract and result + +Use the exact F9-SOURCE-DAG-ALL8 backward word retained by PR5720. +Omit its unimplemented physical_add tail. Keep its native word, scalar +program and public boundaries. The source has 112 X fields, 112 Y fields, +408 independent helpers and nine owned fields. Its mode matrix R has eight +rows and satisfies RR^T=56I. + +Each channel holds one fixed scalar combination of the original eight mode +arrays. It pays a canonical ground-to-chart transport of Gaussian rank r. +It can scatter any rational multiples of that payload at any number of +chronological contacts in that chart. It cannot change chart or recompute +its payload from changed data. Different channels can cancel their noise. +Gathering, storage, scalar scatter and all helper restoration are granted. +Y restoration is required. These grants enlarge the feasible class. + +For every mixture of ranks 1 through 7, the sum of the paid ranks is at +least 65. There is no bound on the number of channels. Stronger certificates +give rank mass at least 90 for ranks at most 5, and at least 77 for ranks +at most 6. These are lower bounds, not claims of sharp minimum rank mass. + +For H_r=r^q H_1, 0= d_A. + +Indeed, apply pi to the complete matrix sum. A term outside E_A is zero. +Every other channel has matrix rank at most one, irrespective of its +coefficients and of cancellation with other terms. Rank subadditivity +proves the inequality. The full rational span is used for W; no coefficient +search or numerical rank threshold is part of this proof. + +There are also coordinate inequalities. Every row of R^T is nonzero. +For each output coordinate i, at least one used frame must have a response +with nonzero coordinate i. Thus sum_(f whose support contains i) n_f>=1. + +Give nonnegative rational weights to these inequalities. For each allowed +frame f, require the sum of weights on inequalities containing f to be at +most r_f. Then their weighted right sides give a lower bound on +sum_f r_f n_f. The checker rebuilds every S_f, W, d_A and E_A. It checks +every frame load with exact fractions. It does not call an LP solver. + +The retained lower bounds are: + +| Maximum rank | Nonzero charts | Quotient cuts | Exact dual lower bound | Integer consequence | +|---|---:|---:|---:|---:| +| 5 | 561 | 130 | 358138597/4000000 | at least 90 | +| 6 | 707 | 65 | 7637849/100000 | at least 77 | +| 7 | 875 | 45 | 6497157737/100000000 | at least 65 | + +The integer step uses that each paid Gaussian rank and each channel count +is an integer. Zero-response channels cannot improve the correction. +Ground channels have zero Z response and do not enter these bounds. + +## Every concave power tariff + +For 1<=r<=s and 0=r s^(q-1). If the rank mass is at least M, +then the channel price is at least M s^(q-1) H_1. Its ratio to 8H_8 is + + (M/64) (8/s)^(1-q). + +For s=7 and M=65 this is at least 65/64>1 throughout the stated interval. +This is a symbolic comparison for all real q in that interval. It is not +a sampled power check. Every omitted physical cost is nonnegative, so +adding it cannot reverse the exclusion. + +## Why a support cover does not suffice + +The exact support-cover optima at maximum ranks 1 through 6 are +101, 100, 79, 62, 223/4 and 467/9. Each value has a rational feasible +cover and a matching rational dual. PR5755 already supplies the rank-one +value and retains that credit. The new rank-two/rank-three bounds alone +would exclude their mixtures. They do not settle ranks four and higher. + +COVER-LIMIT.json gives an integral rank-four cover of cost 62 on 17 charts. +It covers all 112 required coordinates. Even if each selected chart is +allowed to supply its entire response space for that one charge, five of +the eight target directions are still missing. The exact quotient-rank +cuts detect this failure. Coordinate coverage is not mode delivery. + +## Scope and next boundary + +This is a finite source-bound obstruction for the stated immutable-channel +contract. It uses the original normalized scalar quotient and grants more +helper cleanup than PR5745. It proves neither a native lower bound for +changed words nor a general lower bound for active feedback. It changes no +supplier, compiler, stock, precision/tape, analytic or outer assumption. + +A better fixed-word delivery must use at least one rank-eight or rank-nine +channel together with additional noisy channels. The registered census +includes those frames, but the rank-at-most-seven dual does not price them. +The bounded all-rank search did not resolve that case. Its failure to find +a certificate is not an impossibility result or evidence of a caller. diff --git a/research/active-mixed-rank-obstruction/README.md b/research/active-mixed-rank-obstruction/README.md new file mode 100644 index 0000000000..068fbc92c0 --- /dev/null +++ b/research/active-mixed-rank-obstruction/README.md @@ -0,0 +1,58 @@ +# Mixed ranks one through seven cannot repay the active-mode centers + +Any number of immutable, fixed-chart channels of Gaussian ranks 1 through 7 +needs at least **65 units of rank mass** to repair all eight retained modes. +This includes arbitrary rational payloads, repeated chronological scatter and +joint cancellation of output noise. It grants free helper restoration. +For every power tariff H_r=r^q H_1 with 0= 0, 'Nonnegative support weights') + channel_weight = F(certificate.get('channel_weight', '0')) + minimum_channels = certificate.get('minimum_channels', 0) + need(channel_weight >= 0 and minimum_channels in (0, 9), 'Channel count premise') + loads = {f: sum(weights[i] for i in c['support']) + channel_weight + for f, c in selected.items()} + bound = sum(weights) + minimum_channels * channel_weight + for cut in certificate['cuts']: + weight = F(cut['weight']) + need(weight >= 0 and set(cut['span_frames']) <= selected.keys(), 'Cut domain') + rank, active = rank_cut(selected, target, cut['span_frames']) + need(rank == cut['target_rank'] and + sha256(canonical(active)).hexdigest() == cut['active_sha256'], 'Exact quotient cut') + for f in active: + loads[f] += weight + bound += weight * rank + need(all(loads[f] <= selected[f]['rank'] for f in selected), 'Dual chart capacity') + need(str(bound) == certificate['lower_bound'], 'Exact dual objective') + integer = -(-bound.numerator // bound.denominator) + need(integer == certificate['integer_rank_mass'], 'Integer rank mass') + return dict(rank_cap=cap, minimum_channels=minimum_channels, charts=len(selected), + cuts=len(certificate['cuts']), lower_bound=str(bound), + integer_rank_mass=integer) + + +def support_optimum(charts, certificate): + cap = certificate['rank_cap'] + weights = list(map(F, certificate['weights'])) + counts = {int(f): F(v) for f, v in certificate['counts'].items()} + selected = {f: c for f, c in charts.items() if c['basis'] and 1 <= c['rank'] <= cap} + need(len(weights) == 112 and min(weights) >= 0 and + all(v >= 0 for v in counts.values()) and set(counts) <= selected.keys(), 'Cover domain') + need(all(sum(weights[i] for i in c['support']) <= c['rank'] + for c in selected.values()), 'Cover dual capacity') + need(all(sum(n for f, n in counts.items() if i in selected[f]['support']) >= 1 + for i in range(112)), 'Cover primal feasibility') + value = sum(n * selected[f]['rank'] for f, n in counts.items()) + need(value == sum(weights) == F(certificate['optimum']), 'Exact cover strong duality') + return dict(rank_cap=cap, optimum=str(value)) + + +def fails(action, message): + try: + action() + except ValueError: + return message + raise ValueError('Accepted control: ' + message) + + +def main(write_receipt=False): + for record in json.loads((ROOT / 'ORIGINS.json').read_text()): + raw = (ROOT / record['local']).read_bytes() + need(sha256(raw).hexdigest() == record['sha256'] and + sha1(b'blob ' + str(len(raw)).encode() + b'\0' + raw).hexdigest() == record['git_blob'], + 'Frozen predecessor bytes') + word, modes = load() + charts, target, coverage = scan(word, modes) + saved = json.loads((ROOT / 'CERTIFICATES.json').read_text()) + bounds = [check_certificate(charts, target, c) for c in saved['rank_cuts']] + covers = [support_optimum(charts, c) for c in saved['support_optima']] + # The target has eight independent output directions. + need(len(span(target)) == 8, 'Eight target directions') + intersections = {} + for f, c in charts.items(): + common = len(c['basis']) + 8 - len(span([*c['basis'].values(), *target])) + if common: + intersections[f] = common + need(all(charts[f]['rank'] >= 8 for f in intersections), 'Low-rank clean exclusion') + cover_limit = json.loads((ROOT / 'COVER-LIMIT.json').read_text()) + fields = cover_limit['frames'] + missing, _ = rank_cut(charts, target, fields) + need(missing == cover_limit['missing_target_rank'] == 5 and + sum(charts[f]['rank'] for f in fields) == cover_limit['rank_mass'] == 62 and + set().union(*(charts[f]['support'] for f in fields)) == set(range(112)), + 'Cheap cover misses five target directions') + rank1 = json.loads((ROOT / 'source/PR5755-RANK1.json').read_text()) + # The independent predecessor records every rank-one support. + expected_rank1 = rank1['chart_Z_support'] + need({str(f): sorted(c['support']) for f, c in charts.items() if c['rank'] == 1} + == expected_rank1, 'Independent rank-one supports') + controls = [] + c = saved['rank_cuts'][0] + mutant = deepcopy(c) + mutant['support_weights'] = ['1000'] * 112 + controls.append(fails(lambda: check_certificate(charts, target, mutant), 'overweight dual')) + mutant = deepcopy(c) + mutant['cuts'][0]['target_rank'] += 1 + controls.append(fails(lambda: check_certificate(charts, target, mutant), 'wrong target quotient rank')) + mutant = deepcopy(c) + mutant['cuts'][0]['active_sha256'] = '0' * 64 + controls.append(fails(lambda: check_certificate(charts, target, mutant), 'omitted active chart')) + extra = dict(charts) + extra[-1] = dict(rank=1, basis=span(target), support=set(range(112)), responses=8) + controls.append(fails(lambda: check_certificate(extra, target, c), 'unpriced all-mode chart')) + raw_charts, _, _ = scan(word, modes, restore_y=False) + controls.append(fails(lambda: check_certificate(raw_charts, target, c), 'free final-Y cleanup')) + wrong = deepcopy(word) + op = wrong['programs']['backward'][21113][1] + need(op == ['add', ['L', 51], ['L', 53], 1, 1], 'Main-output control site') + op[3] += op[4] + controls.append(fails(lambda: scan(wrong, modes), 'changed scalar endpoint')) + mutant = deepcopy(saved['support_optima'][0]) + mutant['counts'] = {} + controls.append(fails(lambda: support_optimum(charts, mutant), 'incomplete cover')) + # Three noisy channels can supply two clean directions by cancellation. + a, b, noise = {0:F(1),2:F(1)}, {1:F(1),2:F(-1)}, {2:F(1)} + need(len(span([a,b,noise])) == 3 and + not reduce({0:F(1)}, span([a,b,noise])) and + not reduce({1:F(1)}, span([a,b,noise])), 'Joint noise positive control') + result = dict(coverage=coverage, exact_rank_bounds=bounds, exact_support_optima=covers, + clean_intersections={str(f):d for f,d in intersections.items()}, + controls=controls, cover_limit=cover_limit, scope='Fixed immutable canonical channels; no new kappa') + if write_receipt: + (ROOT / 'RESULT.json').write_text(json.dumps(result, indent=2) + '\n') + else: + need(result == json.loads((ROOT / 'RESULT.json').read_text()), 'Complete receipt') + print('PASS_ACTIVE_MIXED_RANK_OBSTRUCTION', json.dumps(coverage, sort_keys=True)) + + +if __name__ == '__main__': + import argparse + parser = argparse.ArgumentParser() + parser.add_argument('--write-receipt', action='store_true') + main(parser.parse_args().write_receipt) From f7c198d38d5f68b0753dd0a51a2328e145b07638 Mon Sep 17 00:00:00 2001 From: Ali Romman <226877337+aliromman@users.noreply.github.com> Date: Mon, 26 Oct 2026 17:20:33 -0500 Subject: [PATCH 3/3] Record sealed mixed-rank proof verification --- .../active-mixed-rank-obstruction/VALIDATION.md | 15 ++++++++++++--- 1 file changed, 12 insertions(+), 3 deletions(-) diff --git a/research/active-mixed-rank-obstruction/VALIDATION.md b/research/active-mixed-rank-obstruction/VALIDATION.md index f5043b2873..a7b649293d 100644 --- a/research/active-mixed-rank-obstruction/VALIDATION.md +++ b/research/active-mixed-rank-obstruction/VALIDATION.md @@ -14,6 +14,15 @@ weight is checked again. The full native operator, physical implementation, compiler, finite-precision, tape and outer theorem are outside this proof. The common-fixed-vector scalar specialization is the retained premise. -Registered normal/optimized replay, metadata, committed bytes, official -export and sealed Git-free evidence are recorded below before publication. -Independent acceptance and hosted scientific replay are not claimed. +Registered normal and optimized replays pass. Metadata and source pins pass. +Three actual corruptions of retained source files are rejected. Committed-byte +and executable-mode checks pass on scientific commit +266324e1ac2e6a086f718afa07028129cda60c9c. Its official 18-file export passes +integrity and declared-input checks. + +A separate read-only export replay passes with Python -I -O -W error -B, +empty PATH and no Git checkout in 6.92 seconds. All 19 files, including the +export manifest, retain their exact bytes and modes. The final documentation +commit records this evidence; it changes no scientific code, source or witness. +Full-history preflight was not run in this shallow checkout. Independent +acceptance and hosted scientific replay are not claimed.