Construct an all-width dyadic oblique section of the resonant channel

research/resonant-oblique-section/INDEPENDENT.json

@@ -0,0 +1,24 @@
+{
+  "decoder_terms": 1920,
+  "full_width_two_columns": 64,
+  "independent_representation": "packed column action; literal Gaussian integer tensors",
+  "nonhermitian_witness": {
+    "E_entry": [
+      "1/2",
+      "0"
+    ],
+    "E_transposed_entry": [
+      "0",
+      "0"
+    ],
+    "column": 0,
+    "row": 9
+  },
+  "odd_literal_widths": [
+    1,
+    3
+  ],
+  "quarter_phase_mutation_rejected": true,
+  "status": "PASS_INDEPENDENT_OBLIQUE_TENSORS",
+  "width_four_dirty_coordinates": 4096
+}

research/resonant-oblique-section/PROOF.md

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+# A uniform dyadic section with a changed output contract
+
+Let A be exactly the 256-term target coefficient in JoyboyBrian's PR #4797,
+with each actual monomial evaluated columnwise as g_f = g tensor ... tensor g.
+Scalar coefficients are retained outside this tensor power. All fourth-root
+phases, including the actual central word iI, are retained.
+
+## Theorem
+
+The 30-term signed-double-coset recipe in SECTION.json emits a fixed sum B
+of 1,920 free monomials with coefficients in (1/2048) Z. For every positive
+integer f,
+
+    A_f B_f A_f = A_f,        ||B_f||_2 <= 113/32.
+
+Consequently E_f = B_f A_f is an idempotent with exactly the kernel of A_f.
+It retains all information in the source channel: A_f x = A_f y if and only
+if E_f x = E_f y. For every attainable observation z, B_f z is a representative
+with A_f(B_f z)=z. The original observation can remain intact.
+
+E is not required to be the orthogonal projection. The literal width-two
+oracle supplies a non-Hermitian entry witness. Thus this changes the output
+contract of PR #4980; it does not contradict that theorem's necessary primes
+3,5,7 for the canonical orthogonal input projection. No choice of independent
+canonical Gaussian jobs, complete target, or paid saving follows from a section.
+
+## Exact finite certificate
+
+The actual source has equal coefficients on g and (-I)g. This is checked for
+every source term. Therefore A_f=0 at odd f. For even f, work in the finite
+monomial algebra quotient by precisely [-I]=[I]. Normalization subtracts either
+zero or two from all eight phase exponents so that the first is zero or one.
+It does not discard iI. Multiplication is actual monomial composition followed
+by this normalization, so it is associative and respects all even evaluations.
+
+In this quotient the paired coefficients of A are +/-1/8. Let a be the
+lexicographically first quotient monomial, explicitly retained as the recipe
+anchor. Define C = 8 a^-1 A. C has 128 distinct coefficients +/-1.
+
+The verifier derives the full signed left and right stabilizers of C:
+
+    L = {(l,s): l C=s C},       R = {(r,t): C r=t C},
+
+where s,t are +/-1. Their sizes are 32 and 16. This computation is complete:
+if c0 is any support word, a stabilizer must send c0 to another support word,
+so its candidate lies in support(C) c0^-1 on the left, or c0^-1 support(C)
+on the right. Every candidate is tested against the entire coefficient table.
+
+For a recipe representative g, form
+
+    T_g = sum_(r,t in R; l,s in L) t s [r g l].
+
+Equal words are collected. The certificate checks that their nonzero absolute
+multiplicities are one common positive integer m_g. Put U_g=T_g/m_g; U_g
+has coefficients +/-1. With the 30 integer numerators n_g in SECTION.json, set
+
+    D = (1/16384) sum_g n_g U_g.
+
+The emitted D has 1,920 terms, numerators bounded by eight, and the verifier
+checks the complete integer identity
+
+    C (16384 D) C = 16384 C.
+
+Both convolutions are coefficientwise, with no modular rank, floating point,
+sampled tensor width or discovery solver in this certificate. The intermediate
+table (16384 D)C has 5,408 terms; the final table has exactly the 128 prescribed
+terms and their prescribed integer coefficients. It follows that C D C=C.
+
+Define the actual decoder B=8 D a^-1. It follows, in the stated order, that
+
+    A B A = (a C/8) (8 D a^-1) (a C/8)
+          = a C D C /8 = a C/8 = A.
+
+Every emitted monomial is independently checked to have invertible affine
+binary permutation and a mod-four phase polynomial of degree at most two,
+with even quadratic terms. This is the constructive free alphabet used by
+the source: affine address maps, coordinate S phases and conjugate parity-S
+implement the even quadratic terms. No general monomial is silently granted
+free. The even evaluation homomorphism proves the identity at every even f.
+At odd f it holds because A_f=0, irrespective of the chosen lifts of B.
+
+Since B's 1,920 reduced coefficients have common denominator 2048 and integer
+numerators with absolute sum 7,232, their l1 norm is 7,232/2048=113/32.
+Each g_f is unitary, proving the uniform bound. Finally E^2=B(ABA)=BA=E.
+If A x=0 then E x=0; if E x=0 then A x=A E x=0. These implications give
+the kernel equality and exact information equivalence without a rank formula.
+
+## Independent literal checks
+
+independent_tensor.py uses packed column actions, independently reconstructs
+the signed stabilizers and recipe, and applies the resulting actual words
+without a sign quotient when evaluating tensors. It checks every one of the
+64 width-two input columns of ABA=A and E^2=E, and a non-Hermitian entry of E.
+It verifies literal cancellation at odd widths one and three. A dense dirty
+Gaussian-integer input of length 4,096 checks the actual width-four ABA identity.
+These are independent implementation checks, not the universal proof: the
+full finite-algebra identity above is the reason the result holds at all widths.
+
+## Arithmetic and complete obligation ledger
+
+For N=8^f components the displayed decoder uses 1,920 monomial applications
+and at most 1,920 scaled accumulation passes in the inherited linear-array
+interface. Every coefficient is an integer of absolute value at most eight,
+divided by 2048. Its integer numerator needs at most two additions/subtractions
+and three fixed shifts; its denominator is an exact eleven-bit dyadic shift.
+Retain one read-only observation array and use one owned zero accumulator and
+one owned temporary monomial array; initializations and erasures are real
+linear scans. This is an explicit logical buffer plan, not a physical tape
+compiler or a claim that source address applications have unit physical cost.
+
+If real and imaginary parts of the input lie in 2^-P Z and are bounded by M,
+every term and every partial sum can be kept exactly in 2^-(P+11) Z.
+Every emitted scalar has only one nonzero real or imaginary part; hence the
+same bound 113 M/32 < 4M applies componentwise to partial sums. Two extra
+integer magnitude bits suffice, apart from the chosen sign representation.
+The input lattice widening and integer growth are independent of f. Stored
+input errors bounded componentwise by epsilon are amplified by at most
+113 epsilon/32 in each real/imaginary component. No rounding need occur
+inside the decoder at the stated widened grid.
+
+| Obligation | Status |
+|---|---|
+| Exact fixed-word section, all tensor widths, all phases | Proved by full coefficient identity |
+| Same accessible information, changed oblique representative | Proved by kernel equality |
+| Extra paid Gaussian children in this formal linear decoder | Zero |
+| Dyadic scalars, 1,920 logical monomial/accumulation passes | Constructively bounded |
+| Exact input/output grid, uniform magnitude and error factors | Proved above |
+| Logical read-only input plus two owned buffers; initialize/erase scans | Constructively specified |
+| Legal physical reads, routing, tape head motion, precise time invoice | Unknown |
+| Returning arbitrary original owners in a completed paid producer/consumer | Unknown |
+| Encoding useful canonical independent Gaussian jobs in this image | Unknown |
+| Missing complementary target coordinates and eventual buffer funding | Unknown |
+| Source-uniform native compiler, stock, outer contracts and new kappa | Unknown |
+
+The opportunity is removal of odd scalar divisions from this changed recovery
+module. Its 1,920 passes may be too expensive in a completed consumer. This
+does not justify a downstream optimization campaign while that consumer and
+complete invoice remain missing. The family decision is NARROW to this section.
+
+## Discovery and credit
+
+Discovery used the signed-stabilizer double-coset corner of the finite even
+group: 164 nonzero source-corner basis elements and an exact linear system of
+rank 51. Rational elimination gave the 30 dyadic recipe coefficients. The
+discovery system is not needed or trusted by replay; the final integer identity
+directly certifies the resulting section. No random seed or rank extrapolation
+is a premise. Nick Petalas's PR #4825 dyadic width-four minor was an existing
+positive control, not a new contribution of this package.
+
+Credit JoyboyBrian #4797 for the resonant channel, Dugongue #4885 for its
+all-width rank/projector and explicit rational recovery, JoyboyBrian #4980 for
+the orthogonal scalar criterion and explicit oblique open question; retain
+Yiyang/Lyy-iiis #4710, lee-phys #4688, Jacob Alan Sussman #4628 with eumemic
+and Swapnil Jain and the source lineage. Prepared for JoyboyBrian with substantial
+OpenAI Codex assistance. Exact automatic checks are not independent human or
+proof-assistant acceptance. Apache-2.0.

research/resonant-oblique-section/README.md

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+# A dyadic oblique section for the resonant channel
+
+The frozen resonant coefficient A from [PR #4797](https://github.com/CrocSwap/integer-mult-bounds/pull/4797)
+has a fixed dyadic decoder B with **A_f B_f A_f=A_f at every positive tensor
+width**. It uses 1,920 free monomials, denominator 2048 and a uniform operator
+norm upper bound 113/32. No odd scalar division is required in this module.
+
+Given a channel observation z, B_f z is an exact representative of its input
+coset. The idempotent E_f=B_f A_f has exactly the kernel of A_f, so the changed
+representative retains all accessible information. It is genuinely oblique:
+at width two E(9,0)=1/2 while E(0,9)=0. It is not the canonical orthogonal
+projection whose recovery requires primes 3,5,7 in [PR #4980](https://github.com/CrocSwap/integer-mult-bounds/pull/4980).
+That output contract is essential to the predecessor theorem; the new section
+answers its explicit oblique-recovery question without contradicting it.
+
+This is an exact linear interface result, not a complete consumer, restored
+paid supplier, call saving or new kappa. The 1,920 passes may be unaffordable
+in a complete consumer. No selected record or frozen predecessor is modified.
+
+## Certificate and reproduction
+
+[PROOF.md](PROOF.md) derives the result from a finite integer identity.
+[SECTION.json](SECTION.json) has 30 signed-double-coset recipe terms.
+The checker reconstructs the 128-term even target quotient, its signed
+stabilizers (sizes 32 and 16), and the 1,920-term section. It checks the entire
+coefficient table of C D C=C, with A=a C/8 and B=8 D a^-1. Actual central
+quarter phases are retained; only [-I]=[I] is used at even widths. The source
+itself proves odd-width cancellation. All output words are checked against
+the source's affine/quadratic free alphabet.
+
+From the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/resonant-oblique-section
+python3.12 -B scripts/check_research_commit.py --package research/resonant-oblique-section --check-source-history
+```
+
+Direct normal and optimized replay use `verify.py` and
+`independent_tensor.py` in this package. The latter uses a separate packed
+column-action engine: all 64 width-two columns, a non-Hermitian projection
+witness, literal odd-width cancellation and a 4,096-coordinate width-four
+Gaussian-integer dirty input. It rejects loss of actual central quarter phases. Explicit
+checks reject a changed section coefficient, a wrong denominator, loss of the
+central quarter phase and an inadmissible cubic phase. These are deterministic
+exact checks with no solver, network, dependency, Lean or Docker requirement.
+
+The logical decoder retains its observation and uses an owned accumulator and
+one owned monomial temporary. Initialization and erasure scans are required.
+It has 1,920 logical monomial/accumulation passes; every scalar is dyadic with
+eleven extra fractional bits and at most two extra integer magnitude bits.
+The full physical read/routing/tape cost remains unknown. The complete
+[obligation ledger](PROOF.md#arithmetic-and-complete-obligation-ledger) marks
+the missing canonical jobs, complement, original-owner restoration, compiler,
+stock and outer costs explicitly. Unknown costs are not zero.
+
+## Open next questions
+
+1. Can a chronological complete consumer use these changed representatives
+   to encode named independent Gaussian jobs? First bind a canonical encoder,
+   the missing target complement, arbitrary owners and the eventual restored
+   outputs. No useful paid moment is inferred from image rank or this section.
+2. Does the resulting completed consumer afford all 1,920 monomial passes,
+   dyadic arithmetic, two owned buffers, legal reads, routing, initialization,
+   erasure, precision, compiler and outer reserve? Price that complete invoice
+   before extending the mechanism or optimizing the section.
+
+The family decision is **NARROW** to this changed output contract. It removes
+odd division from one explicit formal module. A missing physical consumer is
+not an invitation to treat its cost as zero or to launch a chain of local
+optimizations.
+
+## Sources and credit
+
+Scientific base: canonical main `bee95ce88f9f19802f0613f92838895cd0e14a1f`,
+priority revision 7. The source TARGET is byte-pinned by SHA256 and Git blob in
+[SOURCES.json](SOURCES.json). The selected result remains conditional
+kappa 0.00302015. Overlap at 06:08 UTC covered 310 open PRs (06:05:59),
+4,096 Discussion records (06:06:48), and all 6,690 tracked local notes.
+The oblique match was the predecessor's explicit open question. No novelty
+guarantee follows; arbitrary diffs and unpublished work are outside coverage.
+
+Credit JoyboyBrian #4797 for the channel; Dugongue #4885 for all-width rank,
+orthogonal projection and rational recovery; JoyboyBrian #4980 for the scalar
+criterion and oblique open question; Nick Petalas #4825 for the existing
+width-four dyadic minor. Retain Yiyang/Lyy-iiis #4710, lee-phys #4688,
+Jacob Alan Sussman #4628 with eumemic and Swapnil Jain and the source lineage.
+Signed stabilizer averaging, generalized inverses and oblique projections are
+standard algebra; no literature-priority claim is made. Prepared for
+JoyboyBrian with substantial OpenAI Codex assistance. Exact automated checks
+are not independent human or proof-assistant acceptance. Apache-2.0.

research/resonant-oblique-section/RESULT.json

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+{
+  "D_denominator": 16384,
+  "actual_target_terms": 256,
+  "complete_supplier": false,
+  "controls": [
+    "changed_section_coefficient",
+    "changed_denominator",
+    "retain_quarter_phase",
+    "reject_nonfree_cubic_phase",
+    "reject_boolean_address"
+  ],
+  "decoder_common_denominator": 2048,
+  "decoder_norm_upper_bound": "113/32",
+  "decoder_numerator_histogram": {
+    "-1": 32,
+    "-2": 256,
+    "-3": 128,
+    "-4": 352,
+    "-5": 32,
+    "-6": 64,
+    "-7": 32,
+    "-8": 64,
+    "1": 32,
+    "2": 256,
+    "3": 128,
+    "4": 352,
+    "5": 32,
+    "6": 64,
+    "7": 32,
+    "8": 64
+  },
+  "decoder_terms": 1920,
+  "even_target_terms": 128,
+  "identity": "A_f B_f A_f = A_f for every positive integer f",
+  "left_signed_stabilizer": 32,
+  "new_kappa": false,
+  "nonzero_corner_terms": 30,
+  "oblique_projector_terms": 5408,
+  "operator_claim": "B_f A_f is idempotent with kernel equal to kernel(A_f)",
+  "right_signed_stabilizer": 16,
+  "source_sha256": "e7df852b5c641f326ac67cd35cea8da6cfdd059fa12df1cf6fb758bb4206d684",
+  "status": "PASS_RESONANT_OBLIQUE_SECTION"
+}