diff --git a/research/r480-sharp-charge535/EXPECTED.json b/research/r480-sharp-charge535/EXPECTED.json new file mode 100644 index 0000000000..6f09653ae5 --- /dev/null +++ b/research/r480-sharp-charge535/EXPECTED.json @@ -0,0 +1,9 @@ +{ + "kappa": "49529097302971/31250000000000000", + "kappa_decimal": "0.001584931113695072", + "bit_lower": "793723554003949831983/500000000000000000000000", + "bit_decimal": "0.001587447108007899663966", + "old_kappa": "1584931111785787/1000000000000000000", + "fallback_per_edge": 98682, + "bad_fraction": "1/2305843009213693952" +} diff --git a/research/r480-sharp-charge535/LICENSE b/research/r480-sharp-charge535/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/r480-sharp-charge535/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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Apache-2.0. + +This applies the literal bad-class charge from icekylinx's three-stage cover theorem to the current bit-bound composition. Srikrishnabalaji identified the charge refinement in PR461; PR464 and PR465 applied it to earlier suppliers. This is a transfer of that method, not a newly discovered fallback theorem. + +DreamingOfClouds, with disclosed Claude assistance, supplied PR510's R480 nine-stage word and its exact arithmetic engines; eumemic supplied the R489 source-carrier construction and the original dead-total option. PR535's rank-four complex construction and carryover exposition belong to icekylinx, with GPT-6 Astra and Codex assistance. Original notices are preserved in inputs/. All geometry, compiler, hosting, analytic and original-paper contributions retain their attribution. diff --git a/research/r480-sharp-charge535/README.md b/research/r480-sharp-charge535/README.md new file mode 100644 index 0000000000..1a71dc2b56 --- /dev/null +++ b/research/r480-sharp-charge535/README.md @@ -0,0 +1,39 @@ +# Literal bit bad-class charge on the PR535 composition + +Conditional **kappa = 0.001584931113695072**, versus PR535's **0.001584931111785787**. The absolute gain is only **0.000000000001909285**, about **0.0000001205%**. This is a small arithmetic refinement, not a new circuit or structural breakthrough. + +## What changes + +The bit supplier is PR510's R480 nine-stage word, unchanged. The complex supplier is PR535's rank-four construction, unchanged. Both published histograms and physical invoices remain dependencies. The only changed quantity is the conservative allowance for bad residue classes in the bit moment. + +The pinned source note `inputs/three-stage-cover-bit.tex` proves, for address primes q > 2^80, a bad fraction at most 2m^3/q and a fallback cost at most + +`F = 6N(N-1) + 3N + 6(N-1)`, where `N = 2m`. + +At m = 64, `F = 98682` and `2m^3 / 2^80 = 1/2305843009213693952`. These are strictly smaller than the rounded `32m^2` and `10^-16` allowances still used by PR510. The same observation was applied to earlier suppliers in PR461, PR464 and PR465; those contributions retain credit. + +For the unchanged bit histogram H, E = sum(H_r), live width W, and candidate exponent c, the verified upper moment is + +`sum_r H_r * (r/m)^(1-c) / W + (2m^3/2^80) * F * E * m^(c-1) / W`. + +Good classes are still charged at their full weight; no subtraction of bad classes or splitting of residuals is used. Each bad edge is bounded separately, as in the note. The existing fallback operation and all finite invoice allowances remain conservative because the fallback cost only decreases. The prime threshold is unchanged. + +The exact bit endpoint rises from `0.001587447106092548137108` to **`0.001587447108007899663966`**. It stays below the complex endpoint `0.001589991533606044174031`, so the bit side still binds. Six bootstrap levels and the existing 47-constraint outer assembly give the displayed kappa. + +## Verification + +```sh +python -B verify.py /absolute/path/to/new-output +``` + +Python 3.11+; standard library only; assertions enabled. The output directory must be new and outside this package. + +The verifier reproduces both old component endpoints and the old kappa, certifies the new bit endpoint using two independent rational moment engines, rejects the next 10^-24 bit grid point, and checks all 47 outer inequalities. It rejects the next 10^-18 kappa grid point, invalid assembly parameters, a tampered histogram, and the new bit endpoint under the old rounded allowance. Decimal arithmetic only proposes endpoints; rational intervals decide acceptance. + +The packaged verifier passed on Python 3.11.15 and Python 3.14.8. All mathematical output is identical across versions; only runtime and interpreter metadata differ. Exact receipts and the tested package manifest are preserved in `evidence/`. Both rational engines, all 47 strict outer inequalities, six explicit bootstrap cutoffs and all six negative controls passed. GitHub CI is separately pending; no fresh supplier-operator replay is claimed. + +## Scope + +This retains PR535's conditional adoption of PR510, including the common-label/mutable-source carryover and the source theorem's portability to the compressed nine-stage cover. It does not newly prove that portability. All-size compilation, normalized wrappers, whole-call correlated-helper hosting, restored work, precision/tape and analytic transfer remain conditional. This is not an unconditional multiplication theorem, a full Lean proof, or a measured speedup. + +The inherited suppliers are cited rather than rebuilt by this arithmetic-only verifier. Their source heads, original notices, source theorem and retained receipts are recorded in SOURCES.json and inputs/. The PR535 incremental local replay covered its changed complex construction; it did not rebuild PR510's bit operator. diff --git a/research/r480-sharp-charge535/SOURCES.json b/research/r480-sharp-charge535/SOURCES.json new file mode 100644 index 0000000000..02cf544d3d --- /dev/null +++ b/research/r480-sharp-charge535/SOURCES.json @@ -0,0 +1,23 @@ +{ + "pr510": { + "head": "64edee18af02aab0c1696b677a0287dd0146381b", + "path": "research/e8-r480-nine-stage", + "repo": "DreamingOfClouds/integer-mult-bounds" + }, + "pr535": { + "head": "25f9b7c2310d40c9efdfbe368a40ba6e74d3b0a2", + "numerical_artifacts_unchanged_from": "0d343e2f813f962405481f83d8935299812a7a1c", + "independent_incremental_replay": "Python3.14, fresh reconstruction and changed phases, unchanged phases and bit operator reused" + }, + "source_theorem": { + "repo": "CrocSwap/integer-mult-bounds", + "head": "db0e57bc7f36b5ac0fc844ad7ce1396165ee6647", + "path": "notes/three-stage-cover-bit.tex", + "blob": "f05225d21f92cf8cef21b6e12601d5b4d61991f0" + }, + "sharp_charge_prior_work": { + "pr461": "Srikrishnabalaji", + "pr464": "8c546e9918ae37d842916d89ffe28f2c08cdb3d3", + "pr465": "earlier R715 application; this is a transfer to current bit-bound composition" + } +} diff --git a/research/r480-sharp-charge535/engines/base_two_moment.py b/research/r480-sharp-charge535/engines/base_two_moment.py new file mode 100644 index 0000000000..14a90fa0e7 --- /dev/null +++ b/research/r480-sharp-charge535/engines/base_two_moment.py @@ -0,0 +1,63 @@ +#!/usr/bin/env python3 +"""Maintainer arithmetic cross-check of the pinned PR39 certificate. + +Uses no producer/checker imports. Counts are inputs, not proved by this script. +80-term rational logarithms and 12-term exponentials independently enclose +the two moments. The accompanying written review addresses their realization. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +GRID = 10**30 + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def log_unit(x): + require(1 <= x <= 2, "logarithm range") + z = (x-1)/(x+1) + lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0)) + return lo, lo+2*z**161/(161*(1-z*z)) + + +def log_bounds(x): + k = 0 + while x > 2: + x /= 2 + k += 1 + lo, hi = log_unit(x) + l2, u2 = log_unit(Q(2)) + lo, hi = lo+k*l2, hi+k*u2 + # Outward rational rounding keeps later certificates compact. + return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID) + + +def exp_bounds(x): + require(0 <= x < 1, "exponential range") + term = total = Q(1) + for j in range(1, 13): + term *= x/j + total += term + tail = term*x/13/(1-x/14) + return total, total+tail + + +def moment(m, W, entries, saving): + lo = hi = Q(0) + for t, count in entries: + t, count = int(t), int(count) + require(0 < t < m and count > 0, "improper child") + l, u = log_bounds(Q(m, t)) + lower = exp_bounds(saving*l)[0] + upper = exp_bounds(saving*u)[1] + lo += Q(t*count, m*W)*lower + hi += Q(t*count, m*W)*upper + return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID) + diff --git a/research/r480-sharp-charge535/engines/moment.py b/research/r480-sharp-charge535/engines/moment.py new file mode 100644 index 0000000000..7d697cb452 --- /dev/null +++ b/research/r480-sharp-charge535/engines/moment.py @@ -0,0 +1,113 @@ +"""Independent prospective ordinary five-stage cost, never supplier admission. + +Consumes a raw one-invocation ledger made from the actual source/internal/target +paths. No packed profile is used as an input. Missing physical/finite/compiler +obligations must remain explicit in the accompanying report. +""" +from collections import Counter +from decimal import Decimal, localcontext, ROUND_FLOOR +from fractions import Fraction as F +from functools import lru_cache +from math import factorial +import hashlib, json, sys +from pathlib import Path + +DYAD = 1 << 224 + +def floor(x): return F(x.numerator*DYAD//x.denominator, DYAD) +def ceil(x): return -floor(-x) + +@lru_cache(None) +def logarithm(x): + """Mercator bounds after binary reduction; ln(2) via two Mercator sums. + + For 1 <= y < 2, log(y) is an alternating series in z=y-1. + We instead reduce to 1 <= y < 3/2, and bound log(3/2) and + log(2) by the same alternating series at z=1/2 and z=1/3, + using log(2)=log(3/2)+log(4/3). All sums are exact rationals. + """ + assert x >= 1 + k = 0 + while x >= 2: x /= 2; k += 1 + j = int(x >= F(3, 2)) + if j: x /= F(3, 2) + def mercator(y): + z=y-1 + assert 0 <= z <= F(1,2) + # An even partial sum is below log; the next term bounds its tail. + t=z; s=F(0) + for n in range(1, 225): + s += (t if n & 1 else -t)/n + t *= z + return s, s+t/225 + a,b=mercator(x); c,d=mercator(F(3,2)); e,f=mercator(F(4,3)) + return floor(a+(k+j)*c+k*e), ceil(b+(k+j)*d+k*f) + +def exponential(l,u): + assert 0 <= l <= u < F(1,10) + def poly(x): + s=t=F(1) + for j in range(1,25): t=t*x/j; s+=t + return s,t + lo,_=poly(l);hi,last=poly(u) + return floor(lo),ceil(hi+last*u/25/(1-u/26)) + +def moment(H,m,W,a,bad=False): + lo=hi=F(0) + for r,n in H.items(): + assert 0 < r < m and n > 0 + l,u=logarithm(F(m,r)); e,f=exponential(a*l,a*u) + p=F(r*n,m*W);lo+=p*e;hi+=p*f + if bad: + # Existing envelope shape, only conditionally portable to the new cover. + l,u=logarithm(F(m));e,f=exponential(a*l,a*u) + p=F(32*m*sum(H.values()),10**16*W) + lo+=p*e;hi+=p*f + return floor(lo),ceil(hi) + +def decimal(x): + with localcontext() as c: + c.prec=70 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def certify(H,m,W,bad=False): + with localcontext() as c: + c.prec=70 + terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln()) for r,n in H.items()] + fallback=Decimal(32*m*sum(H.values()))/Decimal(10**16*W) if bad else Decimal(0) + lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003') + for _ in range(210): + a=(lo+hi)/2 + if sum(p*(a*l).exp() for p,l in terms)+fallback*(a*lm).exp()<1:lo=a + else:hi=a + q=int(((lo+hi)/2*10**24).to_integral_value(rounding=ROUND_FLOOR)) + lo=F(q,10**24);hi=lo+F(1,10**24) + lv=moment(H,m,W,lo,bad);hv=moment(H,m,W,hi,bad) + assert lv[1]<10 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0} + assert min(margins.values())==minimum + assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2 + assert 1-eps*(1+c)==eta-eps*eta/4 + return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma, + eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam, + lambda_prime=lp,alpha_squared_power=r,delta=delta, + C0=bridge['semantic']['C0'],C1=1,kappa=kappa), + strict_constraints=slacks,margins=margins,minimum_margin=minimum, + absorption_gap=minimum-kappa, + recurrence=dict(internal=internal,leaf=leaf,reservations=1-c)) diff --git a/research/r480-sharp-charge535/evidence/RESULT-311.json b/research/r480-sharp-charge535/evidence/RESULT-311.json new file mode 100644 index 0000000000..b848db7101 --- /dev/null +++ b/research/r480-sharp-charge535/evidence/RESULT-311.json @@ -0,0 +1,130 @@ +{ + "status": "PASS_LITERAL_BIT_CHARGE_AND_CONDITIONAL_COMPOSITION", + "kappa": "49529097302971/31250000000000000", + "kappa_decimal": "0.001584931113695072", + "old_kappa": "1584931111785787/1000000000000000000", + "absolute_gain": "381857/200000000000000000", + "relative_gain": "1909285/1584931111785787", + "bit_lower": "793723554003949831983/500000000000000000000000", + "bit_decimal": "0.001587447108007899663966", + "old_bit_lower": "396861776523137034277/250000000000000000000000", + "complex_lower": "1589991533606044174031/1000000000000000000000000", + "binding_component": "bit", + "bad_fraction": "1/2305843009213693952", + "fallback_per_bad_edge": 98682, + "sharp_rank_one_mass": "1416136041/36028797018963968", + "old_rank_one_mass": "1836864/152587890625", + "moment_receipts": [ + { + "endpoint": "793723554003949831983/500000000000000000000000", + "accept": true, + "mercator": [ + "-836664777402653340213158591423428800586883162087082187515497/485667223056432267729865476705879726660601709763034880312953102434726071301302124544", + "-1673329554805306680426317182846857601173766252116568920967017/971334446112864535459730953411759453321203419526069760625906204869452142602604249088" + ], + "base_two": [ + "-62067252551835315492669/36028797018963968000000000000000000000000000000", + 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PR510 bit and PR535 complex operators unchanged and cited, not rebuilt. Conditional portability of the source bad-class theorem and all inherited all-size/compiler/realization/hosting/precision/tape/analytic assumptions remain." +} diff --git a/research/r480-sharp-charge535/evidence/RESULT-314.json b/research/r480-sharp-charge535/evidence/RESULT-314.json new file mode 100644 index 0000000000..0f574d3b51 --- /dev/null +++ b/research/r480-sharp-charge535/evidence/RESULT-314.json @@ -0,0 +1,130 @@ +{ + "status": "PASS_LITERAL_BIT_CHARGE_AND_CONDITIONAL_COMPOSITION", + "kappa": "49529097302971/31250000000000000", + "kappa_decimal": "0.001584931113695072", + "old_kappa": "1584931111785787/1000000000000000000", + "absolute_gain": "381857/200000000000000000", + "relative_gain": "1909285/1584931111785787", + "bit_lower": "793723554003949831983/500000000000000000000000", + "bit_decimal": "0.001587447108007899663966", + "old_bit_lower": "396861776523137034277/250000000000000000000000", + "complex_lower": "1589991533606044174031/1000000000000000000000000", + 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PR510 bit and PR535 complex operators unchanged and cited, not rebuilt. Conditional portability of the source bad-class theorem and all inherited all-size/compiler/realization/hosting/precision/tape/analytic assumptions remain." +} diff --git a/research/r480-sharp-charge535/evidence/TESTED-MANIFEST.json b/research/r480-sharp-charge535/evidence/TESTED-MANIFEST.json new file mode 100644 index 0000000000..3eaab2df65 --- /dev/null +++ b/research/r480-sharp-charge535/evidence/TESTED-MANIFEST.json @@ -0,0 +1,19 @@ +{ + "EXPECTED.json": "e2a71c1d49c68c35fb5c4f20f10377d708f55c3d25146fa811a99c7230d6ff0c", + "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4", + "NOTICE.md": "b0e850224252f988736f9700cfef41f44cc19a013a5fb794f6f4e07cc78e09db", + "README.md": "59b6778c98f567cdec84d8bfd940190915a67da6bdc161c234ad107a8be0eaec", + "SOURCES.json": "777159be477270644e59bc19ea5be4f3b09572b42f5679dc23057831c73b2a9d", + "engines/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098", + "engines/moment.py": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482", + "engines/outer.py": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c", + "inputs/PR510-CARRYOVER.md": "b2b597491874d6e4ad3ead49bf9a15515745de578bc8c200ab60cf6ec9bac1f2", + "inputs/PR510-EXPECTED.json": "883f7a1a84f31d9e4430e348bc8d9c1ec10210b206ff602dbe126349056af13d", + "inputs/PR510-NOTICE.md": "f3e97a9876a4d79af9260acc95df9e6b88e46f761a6c4b7a4e75e959468cf47e", + "inputs/PR535-EXACT-ASSEMBLY.json": "246f082c286170d61f0aff3cbc4e4162852e5b162ea67cee274b280c04c4e564", + "inputs/PR535-EXPECTED.json": "cb734055c36681d38cf431a6926b7713351a94073cfba37d70a5384a9ef0d178", + "inputs/PR535-NOTICE.md": "1547136e6d269fdcef2b68c10a97a786a79a4732a12738363f679bfb282430cd", + "inputs/PR535-PREFIX-AND-INVOICE.json": "c79683443afa9a974fe8a8fd009f8efa8358d3f5d8b3972ae058b07869166172", + "inputs/three-stage-cover-bit.tex": "47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511", + "verify.py": "79bd3f0d115a695bb2466449b93b18d521ce4f30d9ea9f3e222ca0b2c2425c13" +} diff --git a/research/r480-sharp-charge535/inputs/PR510-CARRYOVER.md b/research/r480-sharp-charge535/inputs/PR510-CARRYOVER.md new file mode 100644 index 0000000000..da33eb5206 --- /dev/null +++ b/research/r480-sharp-charge535/inputs/PR510-CARRYOVER.md @@ -0,0 +1,125 @@ +# PR510 as a conditional bit dependency: mutable-source carryover + +## Review conclusion + +PR510 at commit **`64edee18af02aab0c1696b677a0287dd0146381b`** may be used as an +explicitly cited conditional bit supplier with + +$$ +c=\frac{396861776523137034277}{250000000000000000000000} + =0.001587447106092548137108. +$$ + +Its nine-stage mutable-source realization is covered by the same role-independent +common-label/mutable-basis contract already retained by PR493 and PR531. The +lemma below supplies the short carryover argument that PR510's README lists +as a review point. It does **not** discharge the inherited all-size compiler, +target-label, whole-call correlated-helper hosting, compressed-window, restored-work, +common-grid/tape, precision, row, or analytic contracts. A claim that the bit +supplier is unconditional would be unsupported. + +The pinned PR510 README and source were reviewed, together with PR472's +“Shared physical labels and residual routes” proof and PR443's replication +proof. No long bit replay was run for this review. + +## Role-independent local realization lemma + +Work over an admitted odd **address** field. Stream **values** are in +$\mathbf F_2$; these two fields are distinct. For an orthogonal projector +$P$, the bit address permutation is + +$$ +D_P=\begin{pmatrix}I-P&P\\P&I-P\end{pmatrix},\qquad D_P^2=I. +$$ + +Let $\Lambda_D f=f\circ D$ denote pullback by an address bijection. Then, +for arbitrary streams $f,g$, + +$$ +\Lambda_D(f\mathbin\oplus g)=\Lambda_Df\mathbin\oplus\Lambda_Dg. +$$ + +Consequently, if both operands carry the same current address label, an XOR +into either operand commutes with that common label. This statement has no +condition on whether its destination is designated a source, helper, or target, +nor on whether the stream still equals its original input. + +For nested frames $F\subseteq G$, their orthogonal projectors satisfy +$P_GP_F=P_F$. Thus $P_G-P_F$ is a projector orthogonal to $P_F$, and + +$$ +D_{P_G-P_F}D_{P_F}=D_{P_G}. +$$ + +This is the usual paid frame climb. Induction through the literal frame climbs +and common-frame XORs therefore gives the same local realization when source +registers are written. The inherited COPY/ERASE semantics for the retained +totals are unchanged. + +In the compressed nine-stage construction, the local projector is extended +by the same exterior stage projector for every register in one invocation. +The exterior summand cancels from nested differences. The explicit port +isometries provide the entry and exit coordinate transports; their conjugacy +is the elementary identity + +$$ +(\rho\oplus\rho)D_P(\rho^{-1}\oplus\rho^{-1})=D_{\rho P\rho^{-1}}. +$$ + +After these specified transports, each common-frame gate uses one common local +chart. Different port transports are not silently identified. The geometry's +endpoint identities, including $D_I D_P=D_{I-P}$, also give the reflected +invocation. Neither induction depends on the number of stages, the replica +number, or the names of its physical source owners. + +## Application to the actual word and owners + +R480 retains PR493's mutable-source structure. Its source writes are the +577 block-D4 XOR gates in phase A. The parent `interface.py` checks that no +target feeds a source/helper, the dirt compensation is target-from-helper at +frame zero, phase B writes only targets, and the full-frame cleanup is the +literal reverse of every source/helper operation. Writing $z=(x,h)$, the +raw map consequently has the triangular form + +$$ +z'=Uz,\qquad y'=y+Jx+Kh. +$$ + +The checked compensation and scalar columns establish the intended target +map; literal $U^{-1}$ restores every source and helper. The preceding +pullback lemma realizes this argument at the recorded frames even while a +source is temporarily changed below the full frame. F2 column replay verifies +the scalar identity; the pullback identities supply the separate address +argument. One is not being substituted for the other. + +The nine-stage `global_k.py` reads both operands' actual frame IDs uniformly, +replays all local and global data/helper columns, and maps the data registers +by the explicit replica/pair/side/port formula. It makes no read-only-source +assumption in that replay. Repetition over the 32 replicas is a direct sum of +the same nine-stage identity. A restored local source may subsequently be a +bridge operand just as in the eleven-stage construction. + +Finally, PR510's `emit_k.py` changes only whole residual owners. Apply the +retained PR442 hosting contract to the **complete arbitrary-dirt local call**; +do not infer a free gate-by-gate identification of different physical families. +The hosting windows exclude every local or bridge touch of their host data +pair. A mutable source in the active pair therefore cannot also be one of that +call's idle hosts. This is the same owner/window obligation as before. + +## Provenance and limited review scope + +- [PR510 README, pinned](https://github.com/DreamingOfClouds/integer-mult-bounds/blob/64edee18af02aab0c1696b677a0287dd0146381b/research/e8-r480-nine-stage/README.md): exact bit endpoint, stage count, declared interfaces and review point. +- [PR493 README as vendored at that pin](https://github.com/DreamingOfClouds/integer-mult-bounds/blob/64edee18af02aab0c1696b677a0287dd0146381b/research/e8-bit-carrier-skeleton/README.md), and its `code/bit/independent/interface.py`: mutable-source word and structural guards. +- [PR472 proof, pinned](https://github.com/chafreaky/integer-mult-bounds/blob/259c68488b7aa8a14cf3fcae7fc499381c9bf729/research/e8-r715-whole-hosted/PROOF.md): role-independent common-label pullback and whole-call hosting scope. +- [PR443 proof, pinned](https://github.com/chafreaky/integer-mult-bounds/blob/5a514e94f156edcc5fd827b87494d962f79d0f6c/research/e8-r740-replica32/PROOF.md): direct-sum replica construction and conditional realization scope. + +The source review also covered PR510's vendored `global_k.py`, `emit_k.py`, +and the parent `code/nine/geometry.py`. This review does not re-certify the +published bit price or all hosting assignments; those remain pinned dependency +evidence to be bound by the combined package. + +Credit: DreamingOfClouds and disclosed Claude assistance for PR510 and its +nine-stage integration; eumemic for PR493; the attributed PR503 R480 change; +Zhihao Chen for PR425's mutable-basis lemma and PR386 geometry; chafreaky for +PR443/444/472 and their disclosed assistance; DaysSky for PR442 hosting. +This carryover exposition was prepared for icekylinx with GPT-6 Astra assistance. diff --git a/research/r480-sharp-charge535/inputs/PR510-EXPECTED.json b/research/r480-sharp-charge535/inputs/PR510-EXPECTED.json new file mode 100644 index 0000000000..c89fe719e8 --- /dev/null +++ b/research/r480-sharp-charge535/inputs/PR510-EXPECTED.json @@ -0,0 +1,102 @@ +{ + "nine": { + "bit_lower": "396861776523137034277/250000000000000000000000", + "bit_lower_decimal": "0.001587447106092548137108", + "profile": { + "1": 345600, + "2": 430080, + "4": 67200, + "6": 49920, + "7": 23040, + "8": 2592 + }, + "bit": { + "live": 30720, + "physical": 30729, + "maxchild": 8, + "rank_mass": 1956096, + "hosted_residuals": 138240, + "bank_assignments": 0, + "prefix_bits": 193, + "charged_prefix_bits": 256, + "coefficient_bits": 193, + "two_level_coefficient_bits": 425, + "charged_two_level_coefficient_bits": 512, + "row_coefficient": 18355, + "halving_depth": 2 + }, + "kappa": { + "pr484": { + "kappa": "662515748707749/500000000000000000", + "kappa_decimal": "0.001325031497415498", + "binding_component": "complex", + "complex_lower": "663394767670133/500000000000000000" + }, + "pr471": { + "kappa": "260706019542937/200000000000000000", + "kappa_decimal": "0.001303530097714685", + "binding_component": "complex", + "complex_lower": "1305231506267591/1000000000000000000" + }, + "pr458": { + "kappa": "323880830669511/250000000000000000", + "kappa_decimal": "0.001295523322678044", + "binding_component": "complex", + "complex_lower": "1297203880559577927527/1000000000000000000000000" + } + }, + "nine_stage_payload_bits": 102, + "stages": 9, + "replicas": 32, + "m": 64 + }, + "parent_r480": { + "pr": 503, + "bit_lower": "393688338238452162729/250000000000000000000000", + "bit_lower_decimal": "0.001574753352953808650916", + "stages": 11 + }, + "parent_r489": { + "pr": 493, + "bit_lower": "391360506745414227149/250000000000000000000000", + "bit_lower_decimal": "0.001565442026981656908596", + "stages": 11 + }, + "findings": { + "lp_helpers_per_4frame": 16, + "lp_total": 480, + "levels": { + "1": { + "frames": 120, + "helper_visits": 240 + }, + "2": { + "frames": 180, + "helper_visits": 480 + }, + "4": { + "frames": 60, + "helper_visits": 360 + }, + "6": { + "frames": 180, + "helper_visits": 480 + }, + "7": { + "frames": 120, + "helper_visits": 240 + } + }, + "cells": { + "2,2,0": 60, + "3,0,0": 120 + }, + "d6": { + "2,1,2": 60, + "3,0,0": 120 + }, + "pivots_max": 0, + "line_classes": 120, + "class_nullity": 0 + } +} \ No newline at end of file diff --git a/research/r480-sharp-charge535/inputs/PR510-NOTICE.md b/research/r480-sharp-charge535/inputs/PR510-NOTICE.md new file mode 100644 index 0000000000..7b01935825 --- /dev/null +++ b/research/r480-sharp-charge535/inputs/PR510-NOTICE.md @@ -0,0 +1,32 @@ +# Notice + +Apache-2.0 (`LICENSE`). Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. + +## New in this package + +- `code/assemble9.py`: the nine-stage bit invoice of PR #463's `assemble_k.py` (verbatim; its complex half is dropped) + followed by PR #493's per-supplier outer-assembly loop (verbatim). +- `code/adapt.py`: writes the adapted `audit_parent_k.py` (2 listed edits, each anchor checked to occur exactly once). +- `data/selection9-r480.json`: the nine-stage hosting selection, nominated by PR #442's programme through the vendored + `emit_k.py --discover` and rediscovered by `verify.py`. +- `code/findings/levels.py` (frame-level register visits). `code/findings/flowlp2.py` (the junk-dimension min-flow LP) and + `code/findings/piv0.py` (pivot classes) are byte-identical copies of DreamingOfClouds research scripts (Claude Sonnet 5.5 / + Claude Opus 5.5 via Claude Code), not from a PR; sha256 in `SOURCE.json`. +- `verify.py`, `EXPECTED.json`, `MANIFEST.json`, `SOURCE.json`, `README.md`. + +## Vendored, byte-identical (`vendor/`; git blob ids and sha256 in `SOURCE.json`, recomputed by `verify.py`) + +From PR #463 (DreamingOfClouds, branch e8-carrier-rekid-r715, head cdf0c1c6540df987467988fd771340670f4136f3): +- `research/e8-r715-eleven-stage-whole/code/{global_k,emit_k,audit_parent_k,audit_hosted_k,hosting_k,bank_routes_k}.py` + (DreamingOfClouds with Claude Opus 5.5, derived from PR #443 and PR #444 by chafreaky with OpenAI Codex, PR #442 by + DaysSky with Claude, and PR #386 by Zhihao Chen with OpenAI Codex; lineage per file in `SOURCE.json`); +- `research/e8-r740-replica32/code/{config,bank_routes,tile_m,e8core}.py` and `engines/{moment,base_two_moment,outer}.py` + (PR #443, chafreaky with OpenAI Codex; the engines are PR #315's, DreamingOfClouds); +- `research/idle-axis-hosting/hosting_core.py` (PR #442, DaysSky with Claude). + +## Used from the parents, unchanged + +- PR #503 (`research/e8-r480-dead-totals`): the R480 word and its verifier, run as the parent. +- PR #493 (`research/e8-bit-carrier-skeleton`, eumemic): the checkers (`columns_k.cpp` and its headers), the residual + isometry record, the nine-stage geometry builder (through the parent run) and the cited complex receipts + (`data/complex-b.json`). 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"c79683443afa9a974fe8a8fd009f8efa8358d3f5d8b3972ae058b07869166172", + "bit_scope": "Pinned PR510 published profile and operator are cited dependencies. Both exact moments are freshly repriced here; the unchanged physical bit operator is not cold-reconstructed. See PR510-MUTABLE-SOURCE-CARRYOVER.md.", + "scope": "Conditional finite witness under the retained all-size compiler, common-label/mutable-basis, whole-call hosting, restored-work, precision/tape and analytic-transfer contracts. Fresh complex admission and finite invoice are separately hash-bound; no unconditional theorem is claimed." +} diff --git a/research/r480-sharp-charge535/inputs/PR535-EXPECTED.json b/research/r480-sharp-charge535/inputs/PR535-EXPECTED.json new file mode 100644 index 0000000000..2c5a1a3262 --- /dev/null +++ b/research/r480-sharp-charge535/inputs/PR535-EXPECTED.json @@ -0,0 +1,62 @@ +{ + "status": "PASS_ASYNC_COMPLEX_AND_CITED_BIT_EXACT_ASSEMBLY", + "kappa": "1584931111785787/1000000000000000000", + "kappa_decimal": "0.001584931111785787", + "complex_lower": "1589991533606044174031/1000000000000000000000000", + "complex_decimal": "0.001589991533606044174031", + "bit_lower": "396861776523137034277/250000000000000000000000", + "bit_decimal": "0.001587447106092548137108", + "binding_component": "bit", + "improvements": { + "PR531": { + "baseline": "1517760977820331/1000000000000000000", + "relative_gain": "67170133965456/1517760977820331", + "percent_display": "4.42560686" + }, + "rejected_POST208_checkpoint": { + "baseline": "771820390859371/500000000000000000", + "relative_gain": "41290330067045/1543640781718742", + "percent_display": "2.67486649" + }, + "PR533": { + "baseline": "381051987698311/250000000000000000", + "relative_gain": "60723160992543/1524207950793244", + "percent_display": "3.98391577" + } + }, + "global_template_content_sha256": "b9f4b60e56e5153c1fc89370e9ceb09f24b23e1c8ff56f2b47ef9d867aa04f2c", + "raw_word_content_sha256": "b9110e8ddb770a3874c2df2019e04a98c1a25684072f7f8e5772653f59f15db2", + "canonical_word_content_sha256": "7c84ac65c8b37173aa4293934f041915895e8ec30595844cc1534e2a19ae0d25", + "global_profile": { + "1": 158976, + "2": 52864, + "3": 15200, + "4": 20160, + "5": 4032, + "6": 4032, + "7": 352, + "8": 576, + "48": 1632 + }, + "local_profile": { + "1": 2208, + "2": 728, + "3": 184, + "4": 280, + "5": 56, + "6": 56, + "8": 8 + }, + "local_rank_mass": 6016, + "entrance_rank": 376, + "width_inventory": { + "5": 56, + "8": 152, + "9": 296 + }, + "global_prefix_bits": 225, + "live": 7168, + "physical": 7682, + "deficit": 2560, + "scope": "Conditional finite witness under the retained all-size compiler, common-label/mutable-basis, whole-call hosting, restored-work, precision/tape and analytic-transfer contracts. Fresh complex admission and finite invoice are separately hash-bound; no unconditional theorem is claimed." +} diff --git a/research/r480-sharp-charge535/inputs/PR535-NOTICE.md b/research/r480-sharp-charge535/inputs/PR535-NOTICE.md new file mode 100644 index 0000000000..f42e0eed85 --- /dev/null +++ b/research/r480-sharp-charge535/inputs/PR535-NOTICE.md @@ -0,0 +1,70 @@ +# Attribution and dependency notice + +Package: **`integer_mult_round21_rank4_delta`**. + +Copyright 2026 icekylinx. Local review and packaging used OpenAI Codex. + +## New work + +Prepared by **icekylinx with GPT-6 Astra assistance**. The new contribution is +the post-center timing and ordered composition of dirty gauges with rank-four +response kernels, including the 14 disjoint M-helper groups, explicit scalar +word, frame and cleanup treatment, independent local argument, physical global +integration, paid accounting, and combined conditional κ assembly. + +The same helper can first serve its own gauge and then donate to a larger +kernel because compensation occurs before the new pivot dirt is injected. +This ordering, its actual frame nesting, and its global use are the substantive +delta. Earlier gauge/kernel primitives and the SAME504 scalar construction +retain their original authorship. + +## Baseline + +[PR531](https://github.com/CrocSwap/integer-mult-bounds/pull/531), by icekylinx +with the disclosed GPT-6 Astra assistance, supplies the SAME504 construction +and the baseline package `research/direct-delivery504-global`. The final +baseline head is `e07a8a3f9f79c22113398a0c5dbbc7601f489b3e`; the scientific +files used here match the original Round20 pins. + +**The original PR531 NOTICE is retained separately and remains applicable.** +That notice records the scalar, geometry, controller, phase, wrapper, hosting, +and analytic ancestry. This short incremental notice does not replace it. + +In particular, DreamingOfClouds's PR484 response/kernel implementation and its +predecessors retain their source notices. Zhihao Chen's PR425 mutable-basis +lemma, the PR425/433/514 scalar lineage, and the previously credited source +and delivery contributions remain attributed through PR531's notice. + +## Adopted bit supplier + +[PR510](https://github.com/CrocSwap/integer-mult-bounds/pull/510), pinned to +`64edee18af02aab0c1696b677a0287dd0146381b`, is by **DreamingOfClouds with +Claude Opus 5.5 assistance via Claude Code**, as disclosed by that package. +Its R480 nine-stage bit endpoint is adopted as a cited conditional dependency. +It is not a new bit-side research result of this delta, and no cold replay +of that full bit pipeline was performed here. + +**The original PR510 NOTICE is retained separately and remains applicable.** +Its ancestry includes eumemic's PR493 carrier-skeleton word; PR503's R480 +dead-total change; chafreaky's PR443/444 replication pipeline; DaysSky's PR442 +whole-residual hosting and PR381 column checker; Zhihao Chen's PR386 geometry; +DreamingOfClouds's PR315 moment engines and pipeline adaptations; and Dylan +Crow's PR370 framework, with the assistance disclosures in those sources. + +The present PR510 mutable-source carryover exposition is by icekylinx with +GPT-6 Astra assistance. It expands the role-independent common-label argument +already present in PR472 and the mutable-basis lemma of PR425; it does not +claim those underlying ideas as new. + +## Comparison, evidence, and licenses + +PR493 is also retained as the former-bit-supplier control. PR533 is used only +for numerical comparison. A cited rate, reused certificate, or prior notice +does not imply a fresh replay in this package; the README states the completed +validation scope. + +Original source headers, notices, licenses, and pinned provenance remain in +force. This incremental notice does not relicense upstream material. New +Python research/audit files carry the Apache-2.0 notice stated in their source +headers. All reported bounds remain conditional on the listed inherited +interfaces. diff --git a/research/r480-sharp-charge535/inputs/PR535-PREFIX-AND-INVOICE.json b/research/r480-sharp-charge535/inputs/PR535-PREFIX-AND-INVOICE.json new file mode 100644 index 0000000000..b6d6dc05a8 --- /dev/null +++ b/research/r480-sharp-charge535/inputs/PR535-PREFIX-AND-INVOICE.json @@ -0,0 +1,106 @@ +{ + "status": "PASS_INDEPENDENT_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE", + "template_sha256": "b9f4b60e56e5153c1fc89370e9ceb09f24b23e1c8ff56f2b47ef9d867aa04f2c", + "parent_template_sha256": "0cf4e24fa2d87ab9c6808c57221bcbe7217f5d3cc0ce50e51a5be04d03afa307", + "owner_audit_sha256": "86dd8c1851ec818628dee96ceafbfa31611d1a80eb4fd8084b207607795895ba", + "controller_audit_sha256": "e420a2e8d4e861964fdc76127b54e8217808544aa4ba93165da0b1d2aba8d291", + "independent_local_replay_sha256": "e2b567161204bbd2d6eeced1f5550fea904dd714f31c185957f350022d164a23", + "canonical_word_sha256": "e070ae2c78b4a1b644b418774af62b40d8d16e6ba89602f0474afb557d507c37", + "local_programs_sha256": "d403a1b263a43a40448046c7661919206d222bf37e85b63a4a8950144b8c158f", + "local_checks_sha256": "c592e3e4eabbe4c7417546f834c0ee3779d9d139a546d842874726aa193f6997", + "global_prefix_bits": 225, + "charged_prefix_bits": 512, + "complete_invocation_sweeps": 72, + "bridge_sweeps": 72, + "row_prefix_extra": 1024, + "local": { + "backward": { + "events": 22253, + "macros": 17060, + "real_component_primitives": 169716, + "prefix_l1": "1014379823602", + "prefix_bits": 40, + "coefficient_work_primitive_count": 84086, + "all_exact_columns": 728, + "dirty_helper_columns": 504, + "all_dirty_helper_columns_restored": true, + "private_centres_and_coefficient_work_cleared": true, + "pre_wrappers": 208, + "post_wrappers": 208, + "endpoint_evidence": "Hash-bound fresh INDEPENDENT-LOCAL-REPLAY.json" + }, + "forward": { + "events": 22253, + "macros": 17060, + "real_component_primitives": 169668, + "prefix_l1": "86491024246", + "prefix_bits": 37, + "coefficient_work_primitive_count": 84054, + "all_exact_columns": 728, + "dirty_helper_columns": 504, + "all_dirty_helper_columns_restored": true, + "private_centres_and_coefficient_work_cleared": true, + "pre_wrappers": 208, + "post_wrappers": 208, + "endpoint_evidence": "Hash-bound fresh INDEPENDENT-LOCAL-REPLAY.json" + } + }, + "helpers": 504, + "centres": 8, + "live": 7168, + "physical": 7682, + "physical_stock_identity": { + "data_and_owner_families": 7168, + "local_helpers": 504, + "private_centres": 8, + "coefficient_work": 2 + }, + "entering_helpers": 208, + "entrance_rank": 376, + "width_inventory": { + "5": 56, + "8": 152, + "9": 296 + }, + "local_adapters": 345728, + "local_primitive_bound": 8966226286324, + "wrapper_elementary_factors_charged": 4313088, + "factors_per_wrapper": 144, + "compiler": 25934336, + "unhosted_pervertex": 1580469570081368, + "hosted_rebased_pervertex": 1580469569781992, + "conservative_pervertex_used": 4458416559805936, + "retained_prior_overcharge": { + "sha256": "ce1bfa49be88ae67a404f606a11db2232b7e1f2610d4418058e39749d86d8478", + "previous_pervertex_allowance": 2877946989724568, + "added_complete_new_pervertex": 1580469570081368, + "policy": "Retain old allowance and add the entire fresh recount." + }, + "finite": { + "physical_stock_bits": 2641, + "router_bits": 7985, + "literal_bits": 35282, + "physical_row_coefficient": 16467, + "retained_row_coefficient": 20161, + "required_C0_bits": 200012, + "halving_integer_inequality": [ + 5329, + 4608 + ], + "G_exponent": 30000, + "E_exponent": 100000, + "C0_exponent": 210000 + }, + "strict_checks": { + "local_primitive_bound": true, + "logical": true, + "router": true, + "literal": true, + "mass": true, + "precision": true, + "tape": true, + "row": true, + "halving": true + }, + "scope": "Fresh emitted primitive/Horner, prefix, wrapper and finite recount, using the hash-bound independent local all-column/frame replay and actual whole-owner audit. No old INTEGRATION receipt is used to certify the changed word. The inherited all-size compiler, precision/tape and analytic-transfer contracts remain conditional. Retained payload 512 and physical row allowance 20161 are unchanged." +} diff --git a/research/r480-sharp-charge535/inputs/three-stage-cover-bit.tex b/research/r480-sharp-charge535/inputs/three-stage-cover-bit.tex new file mode 100644 index 0000000000..f05225d21f --- /dev/null +++ b/research/r480-sharp-charge535/inputs/three-stage-cover-bit.tex @@ -0,0 +1,264 @@ +\subsection{Atom contract and common endpoint} +Fix an odd prime $q$, chosen below, and put $R_w=\mathbb Z/q^w\mathbb Z$. +An atom is a complete consecutive $w$-digit address field. Write $C_n$ for +reversal of the $n$ atom positions, without reversing digits inside an atom. +The recursive contract is the involution +\[ + F_A(H,D)=(AC_nD,C_nA^{-1}H), +\] +where $A$ is a diagonal array of units on the head atoms. Its entries may +depend on exposed ancestor row fields and the atom index, but not on the +fresh role fields of this invocation or on the head/data values. Only the +polynomial-time descriptors generated by the construction are required; +no constant-cost oracle for arbitrary functions of a prefix is assumed. +Every call preserves all complete spectator fields and restores its row +coordinate. The original unweighted call has $A=I$. + +For a split idempotent $P$ on the head coordinates set +\[ + D_P=\begin{pmatrix}I-P&PC_n\\C_nP&I-C_nPC_n\end{pmatrix}, + \qquad F=D_I. +\] +These are involutions, and $D_P-I$ has split rank $\operatorname{rank}P$. +Nested commuting-projector identities are polynomial identities and require +no symmetry of $P$. For $n=mf$ and $G=g\otimes I_f$ define +\[ + Q_g=\operatorname{diag}(G,C_nGC_n). +\] +Then $Q_gF=FQ_g$ and $Q_gD_PQ_g^{-1}=D_{GPG^{-1}}$. +Consequently conjugating every base frame by any $g\in\operatorname{GL}_m(R_w)$ +leaves the same full physical endpoint $F$. + +Put $Q_A=\operatorname{diag}(A,I)$. Conjugate the entire completed scalar +word by $Q_A$. Scalar gates mix roles having the same address map, so they +commute with $Q_A$. A current role routing also commutes with $Q_A$, since +it changes only fresh fields. Thus only address residuals change, and the +completed endpoint is $Q_AFQ_A^{-1}=F_A$. More explicitly, when a port +changes charts from $g$ to $gR_s$, its local label changes from $M$ to +$R_s^{-1}MR_s$, and +\[ + (gR_s)(R_s^{-1}MR_s)(gR_s)^{-1}=gMg^{-1}. +\] +This routing is a change of row/chart, not an uncharged dense address map. +The adapted chart $Q_AQ_gQ_A^{-1}$ commutes with $F_A$. None of these +conjugations is executed as an extra normalization pass. + +\subsection{Explicit rational three-stage geometry} +Use the retained PR97 local producer with +\[ + h=23,\qquad v=1771,\qquad R_{\rm loc}=28866,\qquad \ell=506. +\] +Its physical projector ranks and scalar word are unchanged. On +$\mathcal A=\mathbb Q^{23}$ take the form $H_0=(I-J/9)/2$. +For a triple $S$ let $q_S=\chi_S$; then $q_S^tH_0q_S=1$. +Fix $q_0=\chi_{\{1,2,3\}}$ and $K=q_0^\perp$. A rational basis is +\[ + k_i=e_i-(e_i^tH_0q_0)q_0,\qquad 2\le i\le23. +\] +Projection along $q_0$ is injective on the span of $e_2,\ldots,e_{23}$, +since $q_0$ has nonzero first coordinate. Thus this is indeed a basis. +Let $\mathcal B,\mathcal C$ be orthogonal copies of $K$ and put +$\mathcal E=\mathcal A\perp\mathcal B\perp\mathcal C$, of dimension $m=67$. + +Choose a coordinate permutation $\pi_S$ taking $\{1,2,3\}$ to $S$. +It preserves $H_0$. The rational isometry $R_{12,S}$ fixes $q_S$ and +$\mathcal C$, and exchanges $\pi_Sk_i\in q_S^\perp\subset\mathcal A$ +with $k_i\in\mathcal B$. Define $R_{23,S}$ by exchanging the same vectors +with their copies in $\mathcal C$, fixing $q_S$ and $\mathcal B$. +Both maps are involutions. A hyperedge $(g,S)$ uses the invocations +$gR_{12,S},g,gR_{23,S}$, with active spaces +\[ + g(\mathcal B+\mathbb Qq_S),\qquad g\mathcal A, + \qquad g(\mathcal C+\mathbb Qq_S). +\] +Their projectors commute; their pairwise and common intersection is the +projector $P_{gq_S}$, and their sum is $I+2P_{gq_S}$. +This follows from the orthogonal decomposition +$\mathbb Qq_S\perp q_S^\perp\perp\mathcal B\perp\mathcal C$ before +conjugation. These are the cumulative-projector three-stage transfer +identities: the three scalar shears complete the interchange, with zero +interstage connectors and no rank-one endpoint correction. Each port +family visits the full cover bijectively, by right multiplication with +its displayed involution. The physical ledger is therefore three copies +of the local auxiliary, data and center histograms, and three copies of +the exterior histogram $m-\text{nullity}$. + +\subsection{One weighted child on a generic residue class} +Work first on one atom column. Let $g$ be its weighted residual involution, +of split rank $r$ relative to $I$. If the cross block has generic northeast +rank profile, unit lower row and column elimination, without normalizing +pivots, gives +\[ + Lg_{XY}R=ED_pF_0^t, + \quad E=(e_1,\ldots,e_r),\quad + F_0=(e_m,\ldots,e_{m-r+1}), +\] +with invertible diagonal $D_p$. Conjugate by $\operatorname{diag}(L,R^{-1})$. +Split rank and the unit pivot minor give a factorization +\[ + g-I=\binom EB(A_0,D_pF_0^t),\qquad + A_0E+D_pF_0^tB=-2I_r. +\] +For clarity, if $g-I=UV$, then $g^2=I$ gives +$U(VU+2I)V=0$. The unit minor supplies a left inverse of $U$ and a right +inverse of $V$, proving the identity over the local ring itself. +Set +\[ + K=-(B+F_0D_p^{-1})E^t+F_0D_p^{-1}A_0(I-EE^t), + \qquad S=\begin{pmatrix}I&0\\K&I\end{pmatrix}. +\] +The identities $KE=-B-F_0D_p^{-1}$ and +$D_pF_0^tK=A_0+E^t$ give +\[ + SgS^{-1}=\begin{pmatrix} + I-EE^t&ED_pF_0^t\\F_0D_p^{-1}E^t&I-F_0F_0^t + \end{pmatrix}. +\] +This is one weighted reversed interchange on the first $r$ head and last +$r$ tail coordinates. All other factors are unit lower intrabank maps or +lower cross-bank additive maps. Thus variable diagonal units remain in +the child's weight array rather than becoming unsupported routing passes. + +For $n=mf$, group atoms by coarse coordinate $i$ and column $j$. +If $L_P,R_P,D_P$ are unweighted normalizers for the current projector and +$A_j=\operatorname{diag}(A[if+j])$, then +$L_j=A_jL_PA_j^{-1}$ and $R_j=R_P$ are unit lower normalizers for +$A_jPC_m$. The child's exact contiguous index rule is +\[ + A_{\rm child}[kf+j]=A[i_kf+j]D_P[k]. +\] +The cross-bank atom pairing is reversed, and $C_r\otimes C_f=C_{rf}$. +Hence the middle operation is exactly one $rf$-atom call of the same +contract, with no interleaving or gathering. Its coefficients depend only +on parent prefix fields, which become ancestor spectators at the child; +the child's fresh roles come from disjoint unused row stock. + +All unit lower adapters require $O(n)$ atom passes. The retained ordered +affine lemma implements each later-target shear in $O(V)$, including +prefix-controlled offsets. The generated descriptors have polynomial cost +in $w$: good-class evaluation follows one ancestral index per level; +fallback evaluation branches by a constant per level. Since depth is +$O(\log e)$ and $w=\lceil e^{1/1000}\rceil$, this costs $e^{O(1)}=w^{O(1)}$. +Stored ancestor data use $O(w\log e)$ bits. The proof of the ordered affine +lemma still applies with these controls: polynomial preparation per prefix +is absorbed by the complete $q^w$ target fiber. Unrelated spectator counters +are incremented by ripple counting rather than read in their entirety. + +\subsection{Weighted leaves and a genuine local-ring fallback} +At one atom write $J_a=\left(\begin{smallmatrix}0&a\\a^{-1}&0\end{smallmatrix}\right)$, +$U(a)=\left(\begin{smallmatrix}1&a\\0&1\end{smallmatrix}\right)$ and +$L(b)=\left(\begin{smallmatrix}1&0\\b&1\end{smallmatrix}\right)$. +Then +\[ + J_a=\operatorname{diag}(1,-1)U(a)L(-a^{-1})U(a), + \qquad U(a)=J_1L(a)J_1. +\] +Thus a weighted leaf uses four old ordinary atom interchanges, three +prefix-controlled shifts and one fixed sign. With +$a_{\rm old}=384599/10^{10}$ it costs $O(Vw^{1-a_{\rm old}})$. +This constant occurs only at stopped leaves. + +A bad residue class cannot be treated by field Bruhat elimination: +an entry zero modulo $q$ need not vanish modulo $q^w$. Instead Gaussian +eliminate the full invertible residual of size $N_0=2m$ with unit pivots. +Each invertible Schur complement has a unit in its next pivot column; +row swaps and unnormalized additions clear that column. There are at most +$N_0(N_0-1)$ additions and $N_0$ swaps. Pivot choices are determined by +the unweighted low residue class: diagonal conjugation scales every +Schur-complement row and column by units and preserves its zero pattern. +The final diagonal has product a fixed sign, since the residual has +determinant $(-1)^r$ and row swaps change only the sign. + +A lower addition is a paid additive adapter. An upper addition is +conjugated into a lower addition by two ordinary coarse-field swaps. +For a field of $f$ atoms, let $R_{\rm tail}$ reverse the later field. +Then the ordinary weighted swap is +\[ + J_A=R_{\rm tail}F_AR_{\rm tail},\qquad + J_A(H,D)=(AD,A^{-1}H). +\] +Each $R_{\rm tail}$ is one unweighted reversed call on the first and last +$\lfloor f/2\rfloor$ atoms, fixing a possible middle atom. Thus $J_A$ +costs at most three children of widths $f,\lfloor f/2\rfloor, +\lfloor f/2\rfloor$. A paired diagonal scaling is $J_aJ_1$, at most six +children. The full diagonal is a product of $N_0-1$ such pairs and a fixed +sign. Across the two banks the reversed pairing already makes the swaps +and diagonal pairs directly reversed calls; the preceding ordinary bounds +remain conservative. Reindexing a tail weight by $j\mapsto f-1-j$ changes +only its descriptor. Charging every addition as upper therefore gives +\[ + 6N_0(N_0-1)+3N_0+6(N_0-1)<32m^2. +\] +Every child has width at most $f$, hence ratio at most $1/m$. Coefficients +are computed from constant-sized local-ring matrices and parent weight +descriptors; only units are inverted. This proves the fallback cost, +including its actual reversals, rather than assuming nonunit pivots away. + +\subsection{Finite covers, batching, and the strict moment} +Take $q>2^{80}$ avoiding all fixed rational denominators and rank witnesses. +At atom width $w$ use the finite cover $\operatorname{GL}_m(R_w)$ and its +right-multiplication edges $g\mapsto gR_s$. Every vertex has the unique chart +\[ + g=g_0+qG_1,\quad g_0\in\operatorname{GL}_m(\mathbb F_q), + \quad G_1\in(\mathbb Z/q^{w-1}\mathbb Z)^{m^2}, +\] +using fixed representatives for $g_0$. Right multiplication permutes $g_0$ +and acts affinely on $G_1$, with a carry determined by $g_0$. Its fixed +dimensional rational linear part factors into fixed scalings, ordered +shifts and a constant number of old width-$O(w)$ interchanges. Thus each +fixed port routing costs $O(Vw^{1-a_{\rm old}})$. + +Let $W_0=|\operatorname{GL}_m(\mathbb F_q)|(2v+3R_{\rm loc})$. +The full role count is $W(w)=W_0q^{m^2(w-1)}$. Split preceding rows into +low role type, high matrix and unused stock. Batch \emph{all} high matrices +for each low type in one stream. A frame edge is one child on this whole +batch, of volume $V/W_0$, and routing is executed once per fixed operation +type. Consequently neither the number of tapes nor the number of child +programs grows with $w$. High cardinality cancels in the normalized moment, +but its $m^2(w-1)$ digits are charged in the row reserve. + +For each fixed projector require the leading minors of sizes $1$ through +$r$ to be units. Each cleared minor polynomial in $g_0$ is nonzero: +similarity to $\operatorname{diag}(I_r,0)$ supplies a witness after the +fixed rank is preserved modulo $q$. Their total degree is at most $m^3$. +The polynomial zero bound and conditioning on invertibility (a factor at +most two) give bad fraction at most $2m^3/q<10^{-16}$ for each edge type. +No simultaneous union bound over edge types is needed, since their moments +are averaged separately. Good classes use the whole-rank child; bad +classes use the full $32m^2$-child fallback just proved. + +Let $H_r$ be the three-stage histogram described above and +$W=2v+3R_{\rm loc}=90140$. The retained ledger gives +\[ + \sum_r rH_r=6037356,\qquad Wm-\sum_r rH_r=2024=2v-3\ell, + \qquad \max\{r:H_r>0\}=66. +\] +For $a=747/2000000$, $\tau=1-a$, and $b=10^{-16}$ the conservative moment is +\[ + \mu_\tau\le + \sum_r\frac{H_r}{W}(r/m)^\tau+ + b\frac{32m^2\sum_rH_r}{W}(1/m)^\tau<1. +\] +The entire fallback is added; no displaced good contribution is subtracted. +The exact script \path{scripts/three_stage_cover_network.py} and certificate +\path{certificates/three-stage-cover-network.json}, from the retained axis +\path{certificates/three-stage-cover-bit-input.json}, give +$1-\mu_\tau>6.3397366396\cdot10^{-7}$. +The certificate uses the 32-term positive atanh logarithm series with its +geometric tail, $\exp(u)\le1+u+u^2/(2(1-u/3))$, and upward rounding to +the grid $2^{-120}$. All inputs have $0\le u<3$; these are rational upper +bounds, not floating-point root estimates. Since every child ratio is at +most one and $\tau<1$, this same strict bound implies $\mu_1<1$. + +With $T(n,w)$ denoting time divided by logical volume, the fixed low-type +profile $\{c_r\}$, including fallback classes, satisfies +\[ + T(n,w)\le\sum_rc_rT(r\lfloor n/m\rfloor,w)+Cn+Cw^{1-a_{\rm old}} + \quad(n\ge m),\qquad + T(n,w)\le Cw^{1-a_{\rm old}}\quad(n0 for r,n in p.items()) + assert sum(r*n for r,n in p.items())==bit['bit']['rank_mass'] + assert m*W-sum(r*n for r,n in p.items())==9984 +check_profile(P) +F=6*N*(N-1)+3*N+6*(N-1);delta=Q(2*m**3,2**80);factor=delta*F*E +assert F==e['fallback_per_edge'] and str(delta)==e['bad_fraction'] +assert delta1 and l>1) +def sharp_bounds(c): + lo,hi=M.moment(P,m,W,c,False);fl,fh=M.moment({1:1},m,W,c,False);a,z=B.moment(m,W,list(P.items()),c);g,h=B.moment(m,W,[(1,1)],c);return lo+factor*fl,hi+factor*fh,a+factor*g,z+factor*h +with localcontext()as ctx: + ctx.prec=90;terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln())for r,n in P.items()];fb=Decimal(factor.numerator)/Decimal(factor.denominator)/Decimal(m*W);lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003') + for _ in range(250): + a=(lo+hi)/2 + if sum(p*(a*l).exp()for p,l in terms)+fb*(a*lm).exp()<1:lo=a + else:hi=a + c=Q(int((lo*Decimal(10**24)).to_integral_value(rounding=ROUND_FLOOR)),10**24) +assert c==Q(e['bit_lower']) and c01 and l>1);checks.append(dict(endpoint=str(a),accept=accept,mercator=[str(lo-1),str(hi-1)],base_two=[str(l-1),str(h-1)])) +# Reproduce the baseline assembly, then change only the bit endpoint. +eta=beta=Q(1,10**18);grid=Q(1,10**18);bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},'rows':{'degree_gap':Q(10**6)-Q(51*20161,25)}} +def assemble(bit_c): + s=min(bit_c,(1-beta)*b)-Q(1,10**24);assert 00 + ceil=lambda x:-(-x.numerator//x.denominator) + cut=max(ceil(36/gap**2),ceil(2*(4+bit['bit']['charged_two_level_coefficient_bits'])/gap));assert cut*gap**2>=36 and cut*gap>=2*(4+bit['bit']['charged_two_level_coefficient_bits']);cuts.append(dict(gap=str(gap),cutoff_log2=str(cut)));chain.append(a) + q=chain[-1]*(1-2*eta);margin=(1-eta)*q/(1+q);ticks=margin/grid;k=grid*((ticks.numerator-1)//ticks.denominator);a=Outer.assembly(chain[-1],b,bridge,k,eta=eta,beta=beta);assert len(a['strict_constraints'])==47 and all(v>0 for v in a['strict_constraints'].values());return k,chain,cuts,a +old,_,_,_=assemble(c0);assert old==Q(e['old_kappa']) +k,chain,cuts,a=assemble(c);assert k==Q(e['kappa']) and k>old +controls=[] +for name,kk,et,bt in [('next_grid',k+grid,eta,beta),('zero_eta',k,Q(0),beta),('zero_beta',k,eta,Q(0)),('unit_beta',k,eta,Q(1))]: + try:Outer.assembly(chain[-1],b,bridge,kk,eta=et,beta=bt) + except AssertionError:controls.append(name) + else:raise AssertionError(name) +mut=dict(P);mut[1]+=1 +try:check_profile(mut) +except AssertionError:controls.append('tampered_bit_histogram') +else:raise AssertionError('tampered_bit_histogram') +# The new endpoint must fail under the old rounded charge, establishing that this is the actual source of gain. +lo,hi,bl,bh=rounded_bounds(P,m,W,c);assert lo>1 and bl>1;controls.append('new_endpoint_rejected_by_old_charge') +res=dict(status='PASS_LITERAL_BIT_CHARGE_AND_CONDITIONAL_COMPOSITION',kappa=str(k),kappa_decimal=e['kappa_decimal'],old_kappa=str(old),absolute_gain=str(k-old),relative_gain=str(k/old-1),bit_lower=str(c),bit_decimal=e['bit_decimal'],old_bit_lower=str(c0),complex_lower=str(b),binding_component='bit',bad_fraction=str(delta),fallback_per_bad_edge=F,sharp_rank_one_mass=str(factor),old_rank_one_mass=str(Q(32*m*m*E,10**16)),moment_receipts=checks,strict_outer_constraints={n:str(v)for n,v in a['strict_constraints'].items()},bootstrap=cuts,controls=controls,seconds=time.monotonic()-start,python=sys.version,manifest_sha256=sha(R/'MANIFEST.json'),scope='Arithmetic repricing only. PR510 bit and PR535 complex operators unchanged and cited, not rebuilt. Conditional portability of the source bad-class theorem and all inherited all-size/compiler/realization/hosting/precision/tape/analytic assumptions remain.') +(out/'RESULT.json').write_text(json.dumps(res,indent=2)+'\n');print(res['status'],e['kappa_decimal'],flush=True)